Here are some solved problems. Do each problem, covering the answer first, then checking your answer.
PROBLEM 1. Find f′(x) if f(x) = sin(2x^3 ).
Answer: Follow the rule: f′(x) = cos(2x^3 )6x^2
PROBLEM 2. Find f′(x) if f(x) = cos( ).
Answer: f′(x) = =
PROBLEM 3. Find f′(x) if f(x) = tan .
Answer: f′(x) = =
PROBLEM 4. Find f′(x) if f(x) = csc(x^3 + x + 1).
Answer: Follow the rule: f′(x) = − csc(x^3 + x + 1)cot(x^3 + x + 1)(3x^2 + 1)
GRAPHICAL DERIVATIVES
Sometimes, we are given the graph of a function and we are asked to graph the derivative. We do this by
analyzing the sign of derivative at various places on the graph and then sketching a graph of the derivative
from that information. Let’s start with something simple. Suppose we have the graph of y = f(x) below and
we are asked to sketch the graph of its derivative.