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# Derivatives of Exponential and Logarithmic Functions for AP Calculus

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By McGraw-Hill Professional
Updated on Oct 24, 2011

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

### Summary of Derivatives of Exponential and Logarithmic Functions

Let u be a differentiable function of x, then

#### Example 1

y = e3x +5xe3 +e3

#### Example 2

y = xexx2ex

Using the product rule for both terms, you have

y = 3sin x

y = e(x3)

y = (ln x)5

#### Example 6

(Note that ln 5 is a constant. Thus the derivative of ln 5 is 0.)

#### Example 7

y = 2x ln x + x

Using the product rule for the first term,

y = ln(ln x)

y = log5(2x + 1)

#### Example 10

Write an equation of the line tangent to the curve of y = ex at x = 1.

The slope of the tangent to the curve y = ex at x =1 is equivalent to the value of the derivative of y = ex evaluated at x = 1. Using your calculator, enter d(e^(x ), x )|x =1 and obtain e. Thus, m = e, the slope of the tangent to the curve at x =1. At x =1, y = e1 = e, and thus the point on the curve is (1, e ). Therefore, the equation of the tangent is ye = e (x – 1) or y = ex. (See Figure 6.2-1.)

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

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