Notes, Lesson 3.6
Implicit Differentiation

Most of our math work thus far has always allowed us to solve an equation for y in terms of x. When an equation can be solved for y we call it an explicit function. But not all equations can be solved for y. An example is:

This equation cannot be solved for y. When an equation cannot be solved for y, we call it an implicit function. The good news is that we can still differentiate such a function. The technique is called implicit differentiation.

When we implicitly differentiate, we must treat y as a composite function and therefore we must use the chain rule with y terms. The reason for this can be seen in Leibnitz notation: . This notation tells us that we are differentiating with respect to x. Because y is not native to what are differentiating with respect to, we need to regard it as a composite function. As you know, when we differentiate a composite function we must use the chain rule.

Let’s now try to differentiate the implicit function, .

 This is a "folium of Descartes" curve. This would be very difficulty to solve for y, so we will want to use implicit differentiation. Here we show with Leibnitz notation that we are implicitly differentiating both sides of the equation. On the left side we need to individually take the derivative of each term. On the right side we will have to use the product rule. ( ) Here we take the individual derivatives. Note: Where did the y’ come from? Because we are differentiating with respect to x, we need to use the chain rule on the y. Notice that we did use the product rule on the right side. Now we get the y’ terms on the same side of the equation. Now we factor y’ out of the expression on the left side. Now we divide both sides by the  factor and simplify. We can see in a plot of the implicit function that the slope of the tangent line at the point (3,3) does appear to be -1.

Another example: Differentiate:

 Given implicit function Doing implicit differentiation on the function. Note the use of the product rule on the second term We do the algebra to solve for y'. Here we see a portion of plot of the implicit equation with c set equal to 5.. When does it appear that the slope of the tangent line will be zero? It appears to be at about (2.2,2.2). We take our derivative, set it equal to zero, and solve. Now putting x = y in the original implicit equation, we find that... x = y = 2.116343299 We still must use a computer algebra system to solve this cubic equation. The one real answer is shown at the left. This answer does seem consistent with our visual estimate. This can be done in Maple with the following command: >evalf(solve(x^3-x^2-5=0,x));

Links to other explanations of Implicit Differentiation:

World Web Math

University of British Columbia

University of Kentucky's Visual Calculus

Implicit Differentiation Using Maple

S.O.S. Math

University of Califonia - Davis

Karl's Calculus Tutor

Derivatives of Inverse Trigonometric Functions

Thanks to implicit differentiation, we can develop important derivatives that we could not have developed otherwise. The inverse trigonometric functions fall under this category. We will develop and remember the derivatives of the inverse sine and inverse tangent.

 Inverse sine function. This is what inverse sine means. We implicitly differentiate both sides of the equation with respect to x. Because we are differentiating with respect to x, we need to use the chain rule on the left side. We solve the equation for . This is because of the trigonometric identity, . Refer back to the equation in step two above. We have our derivative.

 The inverse tangent function. This is what inverse tangent means. We implicitly differentiate both sides of the equation with respect to x. Because we are differentiating with respect to x, we need to use the chain rule on the left side. We solve the equation for . This is because of the trigonometric identity, . Refer back to the equation in step two above. We have our derivative.

 Check Concepts

 #1: When equations are solved for y, we call them... Choose One Explicit Implicit FunctionsRelations #2: When equations cannot be solved for y, we call them... Choose One Explicit Implicit FunctionsRelations #3: True or False. Implict differentiation helps us to find derivatives of inverse trigonometric functions. Choose One TrueFalse