IMPLICIT DIFFERENTIATION

6. IMPLICIT DIFFERENTIATION

Some examples will show the use of implicit differentiation.

Example 1. Given x + e x = t, find dx/dt and d 2 x/dt 2 .

If we give values to x, find the corresponding t values, and plot x against t, we have a graph whose slope is dx/dt. In other words, x is a function of t even though we cannot solve the equation for x in terms of elementary functions of t. To find dx/dt, we realize that x is a function of t and just differentiate each term of the equation with respect to t (this is called implicit differentiation). We get

Solving for dx/dt, we get

Alternatively, we could use differentials here, and write first dx+e x dx = dt; dividing by dt then gives (6.1).

We can also find higher derivatives by implicit differentiation (but do not use differentials for this since we have not given any meaning to the derivative or dif- ferential of a differential). Let us differentiate each term of (6.1) with respect to t; we get

Solving for d 2 x/dt 2 and substituting the value already found for dx/dt, we get

This problem is even easier if we want only the numerical values of the derivatives at a point. For x = 0 and t = 1, (6.1) gives

and (6.2) gives

2 dt 2 =− . 8 Implicit differentiation is the best method to use in finding slopes of curves with

complicated equations.

Section 7 More Chain Rule 203 Example 2. Find the equation of the tangent line to the curve x 3 − 3y 3 + xy + 21 = 0 at

the point (1, 2).

We differentiate the given equation implicitly with respect to x to get

Substitute x = 1, y = 2:

35 7 Then the equation of the tangent line is

By computer plotting the curve and the tangent line on the same axes, you can check to be sure that the line appears tangent to the curve.

PROBLEMS, SECTION 6

1. If pv a = C (where a and C are constants), find dv/dp and d 2 v/dp 2 .

2. If ye xy = sin x find dy/dx and d 2 y/dx 2 at (0, 0).

3. If x y =y x , find dy/dx at (2, 4).

4. If xe y = ye x , find dy/dx and d 2 y/dx 2

5. If xy 3 − yx 3 = 6 is the equation of a curve, find the slope and the equation of the tangent line at the point (1, 2). Computer plot the curve and the tangent line on the same axes.

6. In Problem 5 find d 2 y/dx 2 at (1, 2).

7. If y 3 −x 2 y = 8 is the equation of a curve, find the slope and the equation of the tangent line at the point (3, −1). Computer plot the curve and the tangent line on the same axes.

8. In Problem 7 find d 2 y/dx 2 at (3, −1).

√ 9. For the curve x 2/3 +y 2/3

√ = 4, find the equations of the tangent lines at (2

2, −2 2), at (8, 0), and at (0, 8). Computer plot the curve and the tangent lines on the same axes.

10. For the curve xe y + ye x = 0, find the equation of the tangent line at the origin. Caution: Substitute x = y = 0 as soon as you have differentiated. Computer plot the curve and the tangent line on the same axes.

11. In Problem 10, find y ′′ at the origin.