CK-12-Calculus

http://www.ck12.org Chapter 6. Transcendental Functions

Example 8:
Evaluate∫tanxdx.
Solution:
To solve, we rewrite the integrand as

∫

tanxdx=

∫ sinx

cosxdx.

Looking at the denominator, its derivative is−sinx. So we need to insert a minus sign in the numerator:

=−

∫ −sinx

cosxdx

=−ln|cosx|+C.

Derivatives of Exponential Functions

We have discussed above that the exponential function is simply the inverse function of the logarithmic function. To
obtain a derivative formula for the exponential function with baseb,we rewritey=bxas

x=logby.

Differentiating implicitly,

1 =yln^1 b·dydx.

Solving fordydxand replacingywithbx,

dy

dx=ylnb=b

xlnb.

Thus the derivative of an exponential function is

d

dx[b

x] =bxlnb.

In the special case where the base isbx=ex,since lne=1 the derivative rule becomes

d

dx[e

x] =ex.

To generalize, ifuis a differentiable function ofx,with the use of the Chain Rule the above derivatives take the
general form