Page 5 - Calculus for the Clueless, Calc II

Chapter 2—
Derivatives of Ex, Ax, Logs., Trig Functions, Etc., Etc.

x
x
We will now take derivatives involving ln x, e , a , f(x) 8(x) , trig functions, and inverse trig functions.
Example 1—

2
Let u = x + 5X + 7. Then y = ln u. So dy/dx = (dy/du)(du/dx) = (1/u)(2x + 5) = (2x + 5)/(x + 5x + 7).
2
Notice, taking derivatives of logs is not difficult. However, you do not want to substitute u = x + 5x + 7. You
2
must do that in your head. If y = ln u, do y' = (1/u)(du/dx) in your head!

Example 2—

The simplest way to do this is to use laws 3, 4, and 5 of the preceding chapter and simplify the expression
2
before we take the derivative. So y = 9 ln (x + 7) + ln (x + 3) - 6 ln x. Therefore

Remember to simplify by multiplying
9(2x) = 18x.

Example 3—

Using law 15, y = ln x/ln 2, where ln 2 is a number (a constant). Therefore

Law 18

u
y = e . y' = e (du/dx). If y = e to the power u, where u = a function of x, the derivative is the original function
u
untouched times the derivative of the exponent.
Example 4—

Law 19

u
y = a . y' = a In a(du/dx). If y = a , the derivative is a (the original function untouched) times the log of the
u
u
u
base times the derivative of the exponent.
Example 5—