Derivatives of Log Functions
Using the graph of y = ln x and the dy/dx option under calculate, complete this table.
x
y = ln x dy/dx
dy/dx as a fraction
2
1
y = ln x
1
2
3
-1
4
-2
5
1
2
3 4
5
6
7
[-1.7,7.7] by [-3.1, 3.1]
Do you see a pattern in relating x and the slope? Use it to determine a formula for the derivative of y = ln x.
d
dx ln x =
Use the chain rule to differentiate:
d
dx ln u =
d
ln(x 2 + 3)
dx
Recall the change of base formula for logarithms: logax =
ln x
ln a
What is the derivative of logau? (Hint: Change the base and then take the derivative. Remember that a is a
constant.)
d
dx logax
Work through these examples:
d
log1 0(3x + 1)
dx
d
log (sin x)
dx 2
d
ln(sec x)
dx
d
dx logau
Logarithmic differentiation means that we rewrite the original function by taking the natural log of both sides,
and then use implicit differentiation.
Try it with this example.
y=
(x 2 + 3)(x + 2)4
3
4 − 2x
ln y = ln
(x 2 + 3)(x + 2)4
3
4 − 2x
Try again with y = ex.
d x
e =
dx
y = e5 x
d u
e =
dx
y = esec x
y = e5 x
2
− 2 x +1
Try again with y = bx. (b is any constant)
d x
b =
dx
y = 5x
d u
b =
dx
2
y = 5 sin x
y = 5 sin
y = x sin x
y = (ln x)2 x +1
x
Now try these:
y = xx