Rule to find the derivative: f′(x) = (ln x) + (log x) . This can be simplified to f′
(x) = .
- f′(x) = 5tan x(sec^2 x) ln 5
The rule for finding the derivative of y = au is = au (ln a) , where u is a function of x.
We get f′(x) = 5tan x (sec^2 x) ln 5.
19.
First, we take the derivative of f(x) : f′(x) = 7x^6 − 10x^4 + 6x^2 . Next, we find the value of x
where f(x) = 1 : x^7 − 2x^5 + 2x^3 = 1. You should be able to tell by inspection that x = 1 is a
solution. Or, if you are permitted, use your calculator. Remember that the AP Exam won’t give
you a problem where it is very difficult to solve for the inverse value of y. If the algebra looks
difficult, look for an obvious solution, such as x = 0 or x = 1 or x = −1.
Now we can use the formula for the derivative of the inverse of f(x): =
, where f(a) = c. This formula means that we find the derivative of the inverse of a
function at a value a by taking the reciprocal of the derivative and plugging in the value of x
that makes y equal to a.
20.
First, we take the derivative of y: . Next, we find the value of x where y =
2: . You should be able to tell by inspection that x = 1 is a solution. Or, if you are
permitted, use your calculator. Remember that the AP Exam won’t give you a problem where it
