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# Procedure for Implicit Differentiation for AP Calculus

based on 2 ratings
By McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Differentiation Practice Problems for AP Calculus

Given an equation containing the variables x and y for which you cannot easily solve for y in terms of x, you can find by doing the following:

Steps     1: Differentiate each term of the equation with respect to x.

1. Move all terms containing to the left side of the equation and all other terms to the right side.
2. Factor out on the left side of the equation.
3. Solve for .

### Example 1

Find if y2 + x2 – 4x = 10.

Step 1:  Differentiate each term of the equation with respect to x. (Note that y is treated as a function of x.)

Step 2:  Move all terms containing to the left side of the equation and all other terms to the right:

Step 3:  Factor out

Step 4:  Solve for

### Example 2

Given x 3 + y 3 =6xy, find

Step 1:  Differentiate each term with respect to x:

Step 2:  Move all terms to the left side:

Step 3:  Factor out

Step 4:  Solve for

### Example 3

Find if (x + y )2 – (xy )2 = x 5 + y 5.

Step 1:  Differentiate each term with respect to x:

Distributing 2(x + y ) and – 2(xy ), you have

Step 2:  Move all terms to the left side:

Step 3:  Factor out

Step 4:  Solve for

### Example 4

Write an equation of the tangent to the curve x 2 + y 2 +19 = 2x +12y at (4, 3). The slope of the tangent to the curve at (4, 3) is equivalent to the derivative at (4, 3).

Using implicit differentiation, you have:

Thus, the equation of the tangent is y – 3 = (1)(x – 4) or y – 3 = x – 4.

### Example 5

Practice problems for these concepts can be found at: Differentiation Practice Problems for AP Calculus

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