Math 201: Derivatives 3.2 Day 14
(1) Use the limits definition of the derivative to compute the derivative of f(x) = x
x−1.
(2) Show that f(x) = x3/5is not differentiable at x= 0. Try to calculate the derivative using
the limit definition at the point x= 0 and see what happens. If this limit does not exist
then the function is not differentiable at x= 0.
(3) Prove that d
dx [f(x) + g(x)] = d
dx f(x)+d
dx g(x). Start by writing down the limit defini-
tion of the left side of the equation. Then try to rearrange it so that it looks like the limit
definition of the right side.
(4) Prove that d
dx cf(x) = cd
dx f(x). Start by writing down the limit definition of the left side of
the equation. Then try to rearrange it so that it looks like the limit definition of the right
side.
(5) Use the derivative rules
d
dxxn=nxn−1d
dx[f(x) + g(x)] = d
dxf(x) + d
dxg(x)d
dxcf(x) = cd
dxf(x)
to compute the following derivatives.
(a) d
dx 3x1000
(b) d
dx [xe+ 3x] (yes the number e)
(c) d
dx [2x−1
−3√x]
(6) The graphs of several functions are given below. On the same axes sketch the derivative of
the functions. Think about when the tangent line has positive slope, negative slope, and
zero slope. Here is an example. The dashed line is the derivative.
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