X Standard form - Ximera X + -> C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/breakGround H Men Custom Suits... BASEXX... O BERHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in an hour Get Help Save Erase Edit Qia Implicit $1 Standard $1.1 Implicit $1.2 $1.3 differentiation form differentiation Derivatives of inverse Two young In this section we exponential mathematicians differentiate functions Abstract. Two young mathematicians discuss the standard form of a line. Check out this dialogue between two calculus students (based on a true story): Devyn Riley, I think we've been too explicit with each other. We should try to be more implicit. Riley I. Um. Don't really. .. Devyn I mean when plotting things! Riley Okay, but I still have no idea what you are talking about. Devyn Remember when we first learned the equation of a line, and the "standard form" was ax + by = c or something, which is totally useless for graphing. Also a circle is 2 2 + y2 = 72 or something, and here y isn't even a function of x. Riley Ah, I'm starting to remember. We can write the same thing in two ways. For example, if you write y = mx + b, then y is explicity a function of x but if you write ax + by = c,X Standard form - Ximera X + -> C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/breakGround H Men Custom Suits... O SHAREXX... @ SHEHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in an hour Get Help Save Erase Edit Qia then y is explicity a function of x but if you write ax + by = c, then y is implicitly a function of x. Devyn What I'm trying to say is that we need to learn how to work with these "implicit" functions. Problem. Consider the unit circle Reveal Hint x2 + y? = 1. The point P = (0, 1) is on this circle. Reason geometrically to determine the slope of the line tangent to x2 + y = 1 at P. The slope is Problem. Consider the unit circle Reveal Hint x ty = 1. The point P = 12,2 is on this circle. Reason geometrically to determine the slope of the line tangent to x2 + y2 = 1 at P. The slope is Previous Next >X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation H Men Custom Suits... O SHARES... @ SHEHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in an hour Get Help Save Erase Edit Qia Implicit $1 Standard $1.1 Implicit $1.2 $1.3 differentiation differentiation Derivatives of form inverse Two young In this section we exponential mathematicians differentiate functions Abstract. In this section we differentiate equations that contain more than one variable on one side. Review of the chain rule Implicit differentiation is really just an application of the chain rule. So recall: Theorem. (Chain Rule) If f(x) and g(x) are differentiable, then d. f(9 (z) ) = f' (9 (x) )9' ( 2 ) . Of particular use in this section is the following. If y is a differentiable function of x and if f is a differentiable function, then dar (f(y)) = f'(3) . dx (4) = f'( ) da' Implicit differentiation The functions we've been dealing with so far have been defined explicitly in terms of the independent variable. For example: x - 2 y = 3x- 2x + 1, y = e 3x y = ac2 - 3x + 2 However, this is not always necessary or even possible to do. Sometimes we choose to or we have to define a function implicitly . In this case, the dependent variable is not stated explicitly in terms of an independent variable. Some examples are:X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation ... H Men Custom Suits... O BABE... 0 9:28x18 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in an hour Get Help Save Erase Edit Qia However, this is not always necessary or even possible to do. Sometimes we choose to or we have to define a function implicitly . In this case, the dependent variable is not stated explicitly in terms of an independent variable. Some examples are: 12 + y2 = 4, 23 +y3 = gxy, 24 + 3x2 = 202/3 + y 2/3 + 1 . Your inclination might be simply to solve each of these equations for y and go merrily on your way. However, this can be difficult or even impossible to do. Since we are often faced with a problem of computing derivatives of such functions, we need a method that will enable us to compute derivatives of implicitly defined functions. We'll start with a basic example. Example. Consider the curve (a circle) defined by: 22 + y2 = 1 (a) Find the slope of the line tangent to the circle at the point (2, 2 ). Explanation.X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... . SHARES... O BERHER($ 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in an hour Get Help Save Erase Edit Qia Explanation. The curve defined by the equation x2 + y? = 1 is not a graph of a function. If we solve for y, we obtain two solutions: y = v1 - x2 andy = -V1 - x2. The point 12, 12 ) lies on the graph of the function f(x) = 1 - x2. Let's compute the derivative of f. f'(2) = 2V1 -22 Therefore, the slope of the tangent line at the point (2, 2 ) is given byX Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... O BABEXX... O SHERHAS 20... b Big Dude USA - Bi... >0 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in an hour Get Help Save Erase Edit Qia Therefore, the slope of the tangent line at the point (2, 2 ) is given by slope = f' 22 ? (b) Find the slope of the line tangent to the circle at the point (2, -2 ). Explanation. y = - V1 - x2 The curve defined by the equation x2 + y2 = 1 is not a graph of a function. If we solve for y, we obtain two solutions: y = v1 - x2 andy = -V1 - x2. The point VZ VZ )X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... . SHARES... O BERHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in an hour Get Help Save Erase Edit Qia The curve defined by the equation x2 + y2 = 1 is not a graph of a function. If we solve for y, we obtain two solutions: y = 1 - x2 andy = -V1 -x2. The point 2, -2lies on the graph of the function f(x) = -V1 - 12 . Let's compute the derivative of f. ? f'(a) = 2V1 -22 Therefore, the slope of the tangent line at the point (2, -12 ) is given by slope = f' (2 2 ) ? Notice that we had to differentiate twice, not to mention that we had to first solve for y in terms of x in order to compute these two slopes.X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... O BABE.. @ SHEHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in an hour Get Help Save Erase Edit Qia Let's take a different approach, namely let's use implicit differentiation. Example. Consider the curve (a circle) defined by: x2 +y? = 1 (a) Compute . (b) Find the slope of the line tangent to the circle at (2, 2). (c) Find the slope of the line tangent to the circle at (2, -2). Explanation. The curve defined by the equation x2 + y" = 1 is not a graph of a function. If we solve for y, we obtain two solutions: y = v1 - x2 andy = -V1 - x2. Therefore, we can say that any point (x, y) on the curve lies on the graph of some function f. Starting with 2 + yz = 1 we differentiate both sides of the equation with respect to x to obtain do ( 2 + 2 ) = -1. dx Applying the sum rule we see 2 2 + d - 21 2 dar = 0. Let's examine each of these terms in turn. To start d -2 2 = da On the other hand, , y2 is somewhat different. Here we assume that y = f(x) for some function f, defined on some open interval (this is true for all points -1 > -10 -5 0 10 powered by desmos Let's see another illustrative example: Example. Consider the curve defined by: 6 4 2X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... BABEXX.. @ BRAHHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in an hour Get Help Save Erase Edit Qia Let's see another illustrative example: Example. Consider the curve defined by: x + y' = 9xy 6 7 4 2 -6 -4 -2 2 4 6 -2 -4 -6 (a) Compute d' (b) Find the slope of the line tangent to this curve at (4, 2). Explanation. Starting withX Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... O BABEXX.. O SHERHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in an hour Get Help Save Erase Edit Qia Explanation. Starting with a ty's = gxy, we differentiate both sides of the equation with respect to x to obtain da ( 2 3 + 23 ) = d gry. da Applying the sum rule we get that d d + d gry. da Let's examine each of these terms in turn. To start d dar On the other hand -y' is somewhat different. Here you assume that y = f(x), on some interval I, and hence by the chain rule do y3 = -to (f ( 20 ) ) 3 = 3(f(2) ) 2 . f'(2) = 3y- Considering the final term 9ry, we assume that y = f(x), on some interval I. Hence da d gry = 9 4 (2 . f(x) ) = 9 (2 . f'(2) + f(2)) dy + gy. Putting this all together we are left with the equation 12 dy 3x2 + 3y- dy dac - + 9y.X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... O SHAREXX... @ SHAHHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in an hour Get Help Save Erase Edit Qia Putting this all together we are left with the equation 31 2 dy - 9x dx 3x2 + 3y dx dy + 9y. At this point, we solve for da . dy . Write 32.2 dy 3x2+ 3y dy + 9y dy 3y dx y - 92 d dy - gy - 3x2 dac dy (372 - 9x) = 9y - 3x2 dy 9y - 3x2 3y - 202 da 3y2 - 9x y2 - 3x For the second part of the problem, we simply plug x = 4 and y = 2 into the formula above, hence the slope of the tangent line at (4, 2) is 2. We've included a plot for your viewing pleasure: 6 9 4 2 -6 -4 -2 4 6 - 2X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... O BASEXX... O BRAHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in an hour Get Help Save Erase Edit Qia For the second part of the problem, we simply plug x = 4 and y = 2 into the formula above, hence the slope of the tangent line at (4, 2) is 2. We've included a plot for your viewing pleasure: 6 4 2 -6 -4 -2 4 6 -2 You might think that the step in which we solve for " could sometimes be difficult. In fact, this never happens. All occurrences an arise from applying the chain rule, and whenever the chain rule is used it deposits a single ~ multiplied by some other expression. Hence our expression is linear in , it will always be possible to group the terms containing da dy together and factor out the dy, e da ) just as in the previous examples. One more last example: Example. Consider the curve defined byX Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... O BRASEXX. @ :2621 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in an hour Get Help Save Erase Edit Qia One more last example: Example. Consider the curve defined by cos (ay) - = 4x23. Compute da . Explanation. First, notice that the problem asks for 45, not . This means the variables have changed places! Not to worry, everything is exactly the same. We apply a to both sides of the equation to get cos(zy) - 2 ) = 27 (4273) which gives us da ? dx y dy x - ydy - 8xy dy 3 d2 + 12x2y. 2 2 Distributing and multiplying by x2 yields dx -x2 y sin(xy) dy x' sin(xy) - x +ydy dy = 823y3 2 + 12 xty?. Grouping terms, factoring, and dividing finally gives us -x2 y sin(zy) dy dx 8x3y dy da + y dy - x3 sin(xy) + + 12xty SO, (y - xysin(xy) - 8x y') dy da = 13 sin(xy ) + x + 12xtyX Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... BASEXX... @ 19:262($ 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in an hour Get Help Save Erase Edit Qia Compute ww . Explanation. First, notice that the problem asks for ", not . This means the variables have changed places! Not to worry, everything is exactly the same. We apply a to both sides of the equation to get cos(zy) - x) = dy (42 which gives us ? dx + x x - y dy y dy 13 dx + 12x2y2. = 8xy dy Distributing and multiplying by x2 yields -xy sin(zy) dy dx _ x3 sin(xy) - x tydy da 37313 d.7 + 12xty. Grouping terms, factoring, and dividing finally gives us xy sin(xy) dy dx da + y dy [ - 8x3 y3 dx = x3 sin(xy) +x+ 12xty SO, (y - xy sin(xy) - 8x3y3) an = 23 sin(xy) + 2 + 12xty and now we see da a3 sin(xy) + + 12xty dy y - x2y sin(xy) - 823y3X Derivatives of inverse exponen X + -> C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInDerivativesOfInverseExponentialFunctions H Men Custom Suits... O SHAREXX.. O SHERHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in an hour Get Help Save Erase Edit Qia Implicit $1 Standard $1.1 Implicit $1.2 $1.3 differentiation form differentiation Derivatives of inverse Two young In this section we exponential mathematicians differentiate functions Abstract. We derive the derivatives of inverse exponential functions using implicit differentiation. Geometrically, there is a close relationship between the plots of et and In(x), they are reflections of each other over the line y = x: 6 : 4 2 -6 -4 2 N 4 6 -2 - In(x) -4X Derivatives of inverse exponen X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInDerivativesOfInverseExponentialFunctions H Men Custom Suits... O BASEXX... O BERHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in an hour Get Help Save Erase Edit Qia 6 : 4 2 -6 -4 2 2 4 6 -2 - In(x) -4 -6 One may suspect that we can use the fact that der = e", to deduce the derivative of In(x). We will use implicit differentiation to exploit this relationship computationally. Theorem. (The Derivative of the Natural Logrithm) d In (x ) =1 . Explanation. Recall In(x) = y exactly when ey = x andX Derivatives of inverse exponen X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInDerivativesOfInverseExponentialFunctions H Men Custom Suits... OSHASEX... @ SHEHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a a DXL Twenty-Eight.. a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in an hour Get Help Save Erase Edit Qia Theorem. (The Derivative of the Natural Logrithm) d In (20 ) = = dx Explanation. Recall In(z) = y exactly when ey = x and Hence es = 0 d a py = - Differentiate both sides. da ,dy es - = 1 Implicit differentiation. da dy 1 dy da py Solve for - da Since y = In(x), d In(x) = Question. Compute: d (- In(cos(2))) = From the derivative of the natural logarithm, we can deduce another fact: Theorem. (The derivative of any logarithm) Let b be a positive real number. ThenX Derivatives of inverse exponen X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInDerivativesOfInverseExponentialFunctions H Men Custom Suits... BASEWA... 0 9:3HR($ 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in an hour Get Help Save Erase Edit Qia From the derivative of the natural logarithm, we can deduce another fact: Theorem. (The derivative of any logarithm) Let b be a positive real number. Then d log6 ( 20 ) = 1 da x In(b ) Explanation. Here we need to remember that In (ac) logb (2 ) = So we may write d log% ( 2 ) = - d In(a) da da 1 d In(x) In(b) dx Question. Compute: d 1087 ( 20 ) = We can also compute the derivative of an arbitrary exponential function.X Derivatives of inverse exponen X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInDerivativesOfInverseExponentialFunctions H Men Custom Suits... . BASEDXA... @ SHAHAR 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA O in an hour Get Help Save Erase Edit Qia We can also compute the derivative of an arbitrary exponential function. Theorem. (The derivative of an exponential function) at = a" . In(a). Explanation. Here we need to be slightly sneaky. Note a" = eln(a") = ex In(a) So we may write d d x In(a) da da = ex In(a). Question. Compute: -72 = Previous Next ->