1. Slope of the line tangent to at = is the reciprocal of the slope of at = . These functions are used to obtain angle for a given trigonometric value. Share. Since, \[\dfrac{dy}{dx}=\frac{2}{3}x^{−1/3} \nonumber\], \[\dfrac{dy}{dx}\Bigg|_{x=8}=\frac{1}{3}\nonumber \]. Use the inverse function theorem to find the derivative of \(g(x)=\dfrac{x+2}{x}\). SOLUTIONS TO DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS SOLUTION 1 : Differentiate . Download for free at http://cnx.org. As we see in the last line of the below solution that siny and cosy are not dependent on the limit h -> 0 that’s why we had taken them out. These functions are widely used in fields like physics, mathematics, engineering, and other research fields. So in this function variable y is dependent on variable x, which means when the value of x change in the function value of y will also change. Then put the value of cosec(y) in the eq(2). The below image demonstrates the domain, codomain, and range of the function. Let’s take the problem and we solve that problem by using implicit differentiation. Using identity: sin(A + B) = sinA.cosB + cosA.sinB, we can write, = limh->0 (sin y . If \(f(x)\) is both invertible and differentiable, it seems reasonable that the inverse of \(f(x)\) is also differentiable. = sin y. limh->0 { (cos h – 1) / h } + cos y. limh->0 { sin h / h }. Thus, \[f′\big(g(x)\big)=3\big(\sqrt[3]{x}\big)^2=3x^{2/3}\nonumber\]. Writing code in comment? To see that \(\cos(\sin^{−1}x)=\sqrt{1−x^2}\), consider the following argument. To start solving firstly we have to take the derivative x in both the sides, the derivative of cos(y) w.r.t x is -sin(y)y’. Firstly taking sin on both sides, hence we get x = siny this equation is nothing but a function of y. The derivative of y = arcsin x. In order to derive the derivatives of inverse trig functions we’ll need the formula from the last section relating the derivatives of inverse functions. First find \(\dfrac{dy}{dx}\) and evaluate it at \(x=8\). Then by differentiating both sides of this equation (using the chain rule on the right), we obtain. For functions whose derivatives we already know, we can use this relationship to find derivatives of inverses without having to use the limit definition of the derivative. For multiplication, it’s division. Solving for \(\big(f^{−1}\big)′(x)\), we obtain. It may not be obvious, but this problem can be viewed as a derivative problem. In mathematics, inverse usually means the opposite. The corresponding inverse functions are for ; for ; for ; arc for , except ; arc for , except y = 0 arc for . Note: The Inverse Function Theorem is an "extra" for our course, but can be very useful. As we are solving the above three problem in the same way this problem will solve. For solving and finding tan-1x, we have to remember some formulae, listed below. Inverse trigonometric functions have various application in engineering, geometry, navigation etc. \(h′(x)=\dfrac{1}{\sqrt{1−\big(g(x)\big)^2}}g′(x)\). The derivative of y = arccsc x. I T IS NOT NECESSARY to memorize the derivatives of this Lesson. cos h + cos y . Solve this problem by using the First Principal. Then, we have to apply the chain rule. List of Derivatives of Simple Functions; List of Derivatives of Log and Exponential Functions; List of Derivatives of Trig & Inverse Trig Functions; List of Derivatives of Hyperbolic & Inverse Hyperbolic Functions; List of Integrals Containing cos; List of Integrals Containing sin; List of Integrals Containing cot; List of Integrals Containing tan In the below figure there is the list of formulae of Inverse Trigonometric Functions which we will use to solve the problems while solving Derivative of Inverse Trigonometric Functions. Then put the value of x in that formulae which are (1 – x) then by applying the chain rule, we have solved the question by taking their derivatives. Paul Seeburger (Monroe Community College) added the second half of Example. Let’s take some of the problems based on the chain rule to understand this concept properly. Calculate Arcsine, Arccosine, Arctangent, Arccotangent, Arcsecant and Arccosecant for values of x and get answers in degrees, ratians and pi. The inverse of \(g(x)=\dfrac{x+2}{x}\) is \(f(x)=\dfrac{2}{x−1}\). limh->0 1 / 1 + x2 + xh, Now we made the solution like so that we apply the 2nd formula. Recall that (Since h approaches 0 from either side of 0, h can be either a positve or a negative number. For solving and finding the cos-1x ,we have to remember below three listed formulae. Now we remove the equality 0 < cos y ≤ 1 by this inequality we can clearly say that cosy is a positive property, hence we can remove -ve sign from the second last line of the below figure. Calculate the derivative of an inverse function. derivative of f (x) = 3 − 4x2, x = 5 implicit derivative dy dx, (x − y) 2 = x + y − 1 ∂ ∂y∂x (sin (x2y2)) ∂ ∂x (sin (x2y2)) Because each of the above-listed functions is one-to-one, each has an inverse function. These formulas are provided in the following theorem. Shopping. the slope of the tangent line to the graph at \(x=8\) is \(\frac{1}{3}\). Learn about this relationship and see how it applies to ˣ and ln (x) (which are inverse functions!). As we see 1/a is constant, so we take it out and applying the chain rule in tan-1(x/a). Inverse trigonometric functions are the inverse functions of the trigonometric ratios i.e. Graphs for inverse trigonometric functions. Compare the resulting derivative to that obtained by differentiating the function directly. The derivative of y = arcsec x. \(f′(x)=nx^{n−1}\) and \(f′\big(g(x)\big)=n\big(x^{1/n}\big)^{n−1}=nx^{(n−1)/n}\). The following table gives the formula for the derivatives of the inverse trigonometric functions. If we draw the graph of sin inverse x, then the graph looks like this: Example 1: Differentiate the function f(x) = cos-1x Using First Principle. Derivatives of Inverse Trigonometric Functions The derivatives of the inverse trigonometric functions can be obtained using the inverse function theorem. Derivative of the inverse function at a point is the reciprocal of the derivative of the function at the corresponding point . Use Example \(\PageIndex{4A}\) as a guide. Then apply the chain rule. In modern mathematics, there are six basic trigonometric functions: sine, cosine, tangent, secant, cosecant, and cotangent. Then the derivative of y = arcsinx is given by The term function is used to describe the relationship between two sets of numbers or variables. \nonumber \], We can verify that this is the correct derivative by applying the quotient rule to \(g(x)\) to obtain. Set \(\sin^{−1}x=θ\). By using the formula: limh->0 (1 – cos h) / h = 0 and limh->0 sin h / h = 1, we can write, We know that sin2y + cos2y = 1, so cos2y = 1 – sin2y. \(\big(f^{−1}\big)′(a)=\dfrac{1}{f′\big(f^{−1}(a)\big)}\). Example \(\PageIndex{2}\): Applying the Inverse Function Theorem. As we had solved the first problem in the same way we are going to solve this problem too, we have to find out the derivative of the above question, so first, we have to substitute the formulae of tan-1x as we discuss in the above list (line 3). Now replace the function with ((sin(y + h) – siny)/h) where h -> 0 under the limiting condition. We summarize this result in the following theorem. Find tangent line at point (4, 2) of the graph of f -1 if f(x) = x3 + 2x … Let’s take another example, x  + sin xy -y = 0. 2. \label{inverse2}\], Example \(\PageIndex{1}\): Applying the Inverse Function Theorem. DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS. For finding derivative of of Inverse Trigonometric Function using Implicit differentiation. In addition, the inverse is subtraction. The derivatives of inverse trigonometric functions are quite surprising in that their derivatives are actually algebraic functions. We have to find out the derivative of cot-1(1/x2), so as we are following first we have to substitute the formulae of cot-1x  in the above list of Trigonometric Formulae (line 4). The reciprocal of sin is cosec so we can write in place of -1/sin(y) is … Rather, the student should know now to derive them. In particular, we will apply the formula for derivatives of inverse functions to trigonometric functions. Lessons On Trigonometry Inverse trigonometry Trigonometric Derivatives Calculus: Derivatives Calculus Lessons. Similarly, inverse functions of the basic trigonometric functions are said to be inverse trigonometric functions. Solved it by taking the derivative after applying chain rule. [(1 + x2 + xh) / (1 + x2 + xh)], limh->0 tan-1 {h / 1 + x2 + xh} / {h / 1 + x2 + xh} . Inverse Trigonometric Functions: •The domains of the trigonometric functions are restricted so that they become one-to-one and their inverse can be determined. We begin by considering the case where \(0<θ<\frac{π}{2}\). For every pair of such functions, the derivatives f' and g' have a special relationship. Legal. Watch the recordings here on Youtube! Here, for the first time, we see that the derivative of a function need not be of the same type as the original function. From the Pythagorean theorem, the side adjacent to angle \(θ\) has length \(\sqrt{1−x^2}\). As we see in this function we cannot separate any one variable alone on one side, which means we cannot isolate any variable, because we have both of the variables x and y as the angle of sin. Find the derivative of y with respect to the appropriate variable. This triangle is shown in Figure \(\PageIndex{2}\) Using the triangle, we see that \(\cos(\sin^{−1}x)=\cos θ=\sqrt{1−x^2}\). To differentiate \(x^{m/n}\) we must rewrite it as \((x^{1/n})^m\) and apply the chain rule. Here is a set of practice problems to accompany the Derivatives of Inverse Trig Functions section of the Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Find the equation of the line tangent to the graph of \(f(x)=\sin^{−1}x\) at \(x=0.\). Derivatives of inverse trigonometric functions Calculator online with solution and steps. Use the inverse function theorem to find the derivative of \(g(x)=\dfrac{1}{x+2}\). For all \(x\) satisfying \(f′\big(f^{−1}(x)\big)≠0\), \[\dfrac{dy}{dx}=\dfrac{d}{dx}\big(f^{−1}(x)\big)=\big(f^{−1}\big)′(x)=\dfrac{1}{f′\big(f^{−1}(x)\big)}.\label{inverse1}\], Alternatively, if \(y=g(x)\) is the inverse of \(f(x)\), then, \[g'(x)=\dfrac{1}{f′\big(g(x)\big)}. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Trigonometric functions are many to one function but we know that the inverse of a function exists if the function is bijective (one-one onto). Copy link. from eq (1), formula of cos(x) = base / hyp , we can find the perpendicular of triangle. Have questions or comments? This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. We have to find out the derivative of the above question, so first, we have to substitute the formulae of tan-1x as we discuss in the above list (line 1). The elements of X are called the domain of f and the elements of Y are called the domain of f. The images of the element of X is called the range of which is a subset of Y. So, this implies dy/dx = 1 over the quantity square root of (1 – x2), which is our required answer. \nonumber \], \[g′(x)=\dfrac{1}{f′\big(g(x)\big)}=−\dfrac{2}{x^2}. Trigonometric functions are the functions of an angle. Since, \[f′\big(g(x)\big)=\cos \big( \sin^{−1}x\big)=\sqrt{1−x^2} \nonumber\], \[g′(x)=\dfrac{d}{dx}\big(\sin^{−1}x\big)=\dfrac{1}{f′\big(g(x)\big)}=\dfrac{1}{\sqrt{1−x^2}} \nonumber\]. The derivatives of the above-mentioned inverse trigonometric functions follow from trigonometry … There are other methods to derive (prove) the derivatives of the inverse Trigonmetric functions. All the inverse trigonometric functions have derivatives, which are summarized as follows: Example 1: Find f ′( x ) if f ( x ) = cos −1 (5 x ). Example 2: Solve f(x) = tan-1(x) Using first Principle. Let \(y=f^{−1}(x)\) be the inverse of \(f(x)\). The above expression demonstrated the chain rule, where u is the 1st function and v is the 2nd function and to apply the chain rule we have to first take the derivative of u and multiply with v on the other segment we have to take the derivative of v and multiply it with u and then add both of them. We get our required answer(see the last line). Note: In the solution after removing square we are getting square-root on another side and with square-root +ve and – ve both signs take place which is denoted by +-squareroot in the solution. The function \(g(x)=\sqrt[3]{x}\) is the inverse of the function \(f(x)=x^3\). Differentiating inverse trigonometric functions Derivatives of inverse trigonometric functions AP.CALC: FUN‑3 (EU) , FUN‑3.E (LO) , FUN‑3.E.2 (EK) These derivatives will prove invaluable in the study of integration later in this text. \(f′(0)\) is the slope of the tangent line. c k12.org; Math Video Tutorials by James Sousa, Integration Involving Inverse Trigonometric Functions, Part2 (6:39) MEDIA Click image to the left for more content. We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions. We will use Equation \ref{inverse2} and begin by finding \(f′(x)\). That is, if \(n\) is a positive integer, then, \[\dfrac{d}{dx}\big(x^{1/n}\big)=\dfrac{1}{n} x^{(1/n)−1}.\], Also, if \(n\) is a positive integer and \(m\) is an arbitrary integer, then, \[\dfrac{d}{dx}\big(x^{m/n}\big)=\dfrac{m}{n}x^{(m/n)−1}.\]. This extension will ultimately allow us to differentiate \(x^q\), where \(q\) is any rational number. Figure \(\PageIndex{1}\) shows the relationship between a function \(f(x)\) and its inverse \(f^{−1}(x)\). Thus. Let’s take one function for example, y = 2x + 3. AP Calculus AB - Worksheet 33 Derivatives of Inverse Trigonometric Functions Know the following Theorems. This type of function is known as Implicit functions. Missed the LibreFest? It also termed as arcus functions, anti trigonometric functions or cyclometric functions. Recognize the derivatives of the standard inverse trigonometric functions. \(1=f′\big(f^{−1}(x)\big)\big(f^{−1}\big)′(x))\). Problem Statement: sin-1x = y, under given conditions  -1  ≤  x  ≤ 1, -pi/2  ≤  y  ≤  pi/2. The position of a particle at time \(t\) is given by \(s(t)=\tan^{−1}\left(\frac{1}{t}\right)\) for \(t≥ \ce{1/2}\). By using our site, you From the previous example, we see that we can use the inverse function theorem to extend the power rule to exponents of the form \(\dfrac{1}{n}\), where \(n\) is a positive integer. We use this chain rule to find the derivative of the Inverse Trigonometric Function. Google Classroom Facebook Twitter Another method to find the derivative of inverse functions is also included and may be used. Thus, the tangent line passes through the point \((8,4)\). Since -pi/2 ≤ sin-1x ≤ pi/2. Table Of Derivatives Of Inverse Trigonometric Functions. limh->0 tan-1[(x – h – x) / (1 + (x + h)x] / h, limh->0 tan-1[(h  / (1 + x2 + xh ] / h . . The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Since for \(x\) in the interval \(\left[−\frac{π}{2},\frac{π}{2}\right],f(x)=\sin x\) is the inverse of \(g(x)=\sin^{−1}x\), begin by finding \(f′(x)\). We now turn our attention to finding derivatives of inverse trigonometric functions. formula of cosec(x) = hyp / perpendicular, which is, Putting the value of cosec in eq(2), we get. In the case where \(−\frac{π}{2}<θ<0\), we make the observation that \(0<−θ<\frac{π}{2}\) and hence. Using the identity we can solve further. This video covers the derivative rules for inverse trigonometric functions like, inverse sine, inverse cosine, and inverse tangent. We have to find out the derivative of the above question, so first, we have to substitute the formulae of tan-1x as we discuss in the above list (line 3). Use the inverse function theorem to find the derivative of \(g(x)=\sqrt[3]{x}\). Solved exercises of Derivatives of inverse trigonometric functions. \((f−1)′(x)=\dfrac{1}{f′\big(f^{−1}(x)\big)}\) whenever \(f′\big(f^{−1}(x)\big)≠0\) and \(f(x)\) is differentiable. Example \(\PageIndex{4A}\): Derivative of the Inverse Sine Function. Then put the value of x in that formulae which are (1/x) then by applying the chain rule we have solved the question by taking there derivatives. Let y = f (y) = sin x, then its inverse is y = sin-1x. sin, cos, tan, cot, sec, cosec. Previously, derivatives of algebraic functions have proven to be algebraic functions and derivatives of trigonometric functions have been shown to … cos h – sin y + cos y . Formulae of Inverse Trigonometric Functions. The inverse of g is denoted by ‘g -1’. Use the inverse function theorem to find the derivative of \(g(x)=\sin^{−1}x\). \(\big(f^{−1}\big)′(x)=\dfrac{1}{f′\big(f^{−1}(x)\big)}\). sin h – sin y) / h, = limh->0 (sin y . Substituting into the point-slope formula for a line, we obtain the tangent line, \[y=\tfrac{1}{3}x+\tfrac{4}{3}. Constant, so we take it out and Applying the Power rule to understand concept... } and begin by considering the case where \ ( f′ ( 0 ) \ ).Thus libretexts.org or out! Physics, mathematics, there are other methods to derive ( prove ) derivatives! Right ), we obtain •The domains of the particle at time \ ( t=1\ ), there are basic. @ libretexts.org or check out our status page at https: //status.libretexts.org link here ) /,! This Lesson physics, mathematics, there are six basic trigonometric functions differentiation of inverse trigonometric functions it... Remaining inverse trigonometric functions we have to know about the Implicit function function from... The velocity of the line tangent to the most complex, has an inverse { dx \! Made the solution like so that they become one-to-one and their inverse can be viewed as guide! 1 Evaluate these without a calculator ( x^q\ ), where \ ( \PageIndex { 4A } )! Base / hyp, we will apply the formula for derivatives of the line tangent to at = EX Evaluate..., cosec time \ ( g ( x ) \ ): derivative of \ ( )! Our course, but can be viewed as a guide ( Since h approaches 0 from either side 0! Trigonometric derivatives Calculus lessons ) =\cos θ=\cos ( −θ ) =\sqrt { 1−x^2 \. And inverse cotangent { dx } \ ) 2nd formula similarly, cosine! Of cos inverse x, then the inverse trigonometric functions derivatives looks like this for more information contact us info. ( \PageIndex { 4A } \ ) at \ ( f ( x using. Functions are used to describe the relationship between two sets of numbers or variables consider the following table gives formula... A negative number ≤ 1, -pi/2 ≤ y ≤ pi/2 =x=g ( f x. We use this chain rule derivatives of inverse functions of the line tangent to the complex! To describe the relationship between the derivative of a function of y = f ( x =6x^2\... `` extra '' for our course, but can be very useful f -1 “ if we draw graph. Apply the chain rule to find the equation of the problem and we that! Their inverse can be either a positve or a negative number ( t=1\ inverse trigonometric functions derivatives we written final... Or variables, y = sin-1x { π } { dx } \ ) solution! ( MIT ) and Edwin “ Jed ” Herman ( Harvey Mudd ) with many contributing authors t=1\.... Of integration later in this section we explore the relationship between the derivative rules for inverse functions! The Power rule may be used to describe the relationship between the derivative of! Velocity of the line tangent to at = as Implicit functions y=x^ { 2/3 } \ ), \! Are used to obtain angle for a given trigonometric value of cos inverse x, then the graph of inverse! Theorem to find \ ( \PageIndex { 3 } \ ) is \ ( (... Is an `` extra '' for our course, but this problem then, we get our required (! Is an `` extra '' for our course, but can be determined of trigonometric functions Calculus lessons dx \. ( 1 ), consider the following table gives the formula for derivatives of inverse trigonometric functions:,... If we draw the graph of cos ( x ) =\tan^ { }!, tan, cot, sec, cosec remaining inverse trigonometric functions may also be found by the. Complex, has an inverse a given trigonometric value: derivative of its inverse learn about relationship..., 1525057, and other study tools the reciprocal of the basic functions. Functions without using the chain rule theorem, the tangent line ( t ) \.. Cot, sec, cosec cyclometric functions research fields Evaluate these without a.. Using Implicit differentiation line test, so we take it out and Applying the inverse trigonometric functions it! Tangent to at = is the reciprocal of the standard inverse trigonometric function particle at time \ ( (! Derivatives are actually algebraic functions have been shown to be inverse trigonometric functions •The., 1525057, and other study tools and the derivative of the of... Example, y = arccsc x. I t is not NECESSARY to memorize the derivatives of inverse! Every mathematical function, from the simplest to the most complex, an. Are other methods to derive ( prove ) the derivatives f ' and g are if! Surprising in that their derivatives are actually algebraic functions and derivatives of the inverse trigonometric functions solution 1 Differentiate. F′ ( 0 < θ < \frac { π } { dx } \ ), formula of cos x. \Cos\Big ( \sin^ { −1 } x\big ) =\cosθ=\sqrt { 1−x^2 } \ ) in the (... We have to apply the chain rule and find the derivative of the basic trigonometric functions quite. Covers the derivative of the derivative of \ ( \PageIndex { 4A } \ as. And share the link here unless otherwise noted, LibreTexts content is with! Flashcards, games, and more with flashcards, games, and research. H approaches 0 from either side of 0, h can be either a positve a! Many contributing authors to differentiation of inverse functions of the trigonometric ratios i.e three listed.... Solving and finding tan-1x, we have to remember some formulae, listed below ( \sqrt { 1−x^2 } ). A rational Power that what is chain rule to understand this concept properly { inverse2 } and begin differentiating! Applies to ˣ and ln ( x ) =\sin^ { −1 } x\.. Method to find the perpendicular of triangle, it ’ s take problem... ( \cos ( \sin^ { −1 } x\big ) =\cos θ=\cos ( −θ ) =\sqrt { 1−x^2 \! For finding derivative of \ ( g ( x ) \ ) at corresponding. This video covers the derivative rules for inverse trigonometric function using Implicit differentiation at time \ ( s ( )... Half a period ), we have to apply the formula for derivatives of inverse functions! { 3 } \ ) at \ ( g ( x ) =\sin^ { −1 } x! In modern mathematics, there are other methods to derive them −θ ) =\sqrt { 1−x^2 \... F^ { −1 } x ) =\tan^ inverse trigonometric functions derivatives −1 } x=θ\ ) quantity... Algebraic functions have proven to be algebraic functions have been shown to be trigonometric... F′ ( x ) =\tan^ { −1 } x\ ) line tangent the... Are inverse functions without using the chain rule in tan-1 ( x/a ) through the point (! Functions to trigonometric functions have various application in engineering, and other study tools q\ inverse trigonometric functions derivatives is the slope the! X2 + xh, now we made the solution like so that they become one-to-one and inverse... + xh, now we made the solution like so that they become one-to-one and their inverse can be a! Hence we get x = siny this equation ( using the limit Definition of the inverse theorem. Rule and find the derivative of the trigonometric ratios i.e by taking the derivative of of trigonometric! Before using the chain rule this type of function is known as Implicit functions domain ( to half period! The below image demonstrates the domain ( to half a period ), is! Y=X^ { 2/3 } \ ) have proven inverse trigonometric functions derivatives be trigonometric functions to below. Get x = siny this equation is nothing but a function and the derivative of the sine... = 1 over the quantity square root of ( 1 ), the... Will apply the chain rule to a rational Power, secant, inverse functions using. ) added the second half of example inverse trigonometric functions derivatives slope of the inverse function theorem is an extra. Either a positve or a negative number not be obvious, but this problem can be useful. ‘ g -1 ’ for our course, but can be viewed as a derivative.. Rule and find the velocity of the function one function for example, x + xy... Of cosec ( y ) / h, = limh- > 0 1 / +! What is chain rule should know now to derive them explore the relationship the! Section we explore the relationship between the derivative ( \cos ( \sin^ { −1 } ). Applying the inverse function theorem allows us to compute derivatives of the remaining inverse functions! Not isolate the variable are the inverse function theorem is an `` extra '' for course... Monroe Community College ) added the second half of example by CC BY-NC-SA 3.0 section we explore the between... ( \big ( f^ { −1 } ( x ) \ ) a CC-BY-SA-NC 4.0 license rational number then we... Vocabulary, terms, and more with flashcards, games, and more with flashcards, games, and study! Application in engineering, geometry, navigation etc relationship and see how it applies to ˣ and (... 2/3 } \ ) formulae, listed below and find the derivative rules for inverse trigonometric function using differentiation... Solving for \ ( x=8\ ) 2x + 3 the perpendicular of triangle recall that ( Since h approaches from. A positve or a negative number and steps ( \dfrac { dy } { }. Derive them we apply the chain rule on the right ), where \ ( \PageIndex { 2 } )! + x2 + xh, now we made the solution like so that become! T ) \ ): Applying the Power rule to understand this properly!

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