IXL - Find derivatives using logarithmic differentiation (Calculus practice)

Find the derivative of the function

Write your answer as a function of

′

ref_doc_title.

Learn with an example

question

Find the derivative of the function

Write your answer as a function of

′

key idea

[

(

)

]

(

)

is a function whose base and exponent depend on

you can find its derivative using logarithmic

differentiation:

Take the natural logarithm of both sides of the equation
y
=
[
g
(
x
)
]
h
(
x
)
.
Use the power property of logarithms to write
ln
(
y
)
=
h
(
x
)
ln
[
g
(
x
)
]
.
Differentiate both sides, using the chain rule to substitute
[
ln
(
y
)
]
′
=
y
′
y
on the left side.
Solve for
y
′
in terms of
y
and
x
.
Substitute
y
=
[
g
(
x
)
]
h
(
x
)
to write
y
′
in terms of
x
.

Common derivatives

u ( x )	u ′ ( x )
ln ( x )	1 x
e x	e x
a x	ln ( a ) a x

solution

The function

has a base and exponent that depend on

so use logarithmic differentiation to find its derivative.

First, take the natural logarithm of both sides. Then, use the power property of logarithms to bring down the exponent on the right-hand side.

(

)

(

)

(

)

(

)

Next, take the derivative of both sides. Use the chain rule on the left-hand side and the product rule on the right-hand side.

[

(

)

]

′

[

(

)

]

′

(

)

′

(

)

[

(

)

]

′

(

)

(

)

′

(

)

Now, solve for

′

and substitute

to write the answer in terms of

′

[

(

)

]

′

[

(

)

]

Looking for more? Try this:

$Confetti$

Excellent!

Now entering the Challenge Zone—are you ready?

Questions
answeredQuestions

Time
elapsedTime

hr	min	sec
00	00	06

SmartScore
out of 100
IXL's SmartScore is a dynamic measure of progress towards mastery, rather than a percentage grade. It tracks your skill level as you tackle progressively more difficult questions. Consistently answer questions correctly to reach excellence (90), or conquer the Challenge Zone to achieve mastery (100)! Learn more

Teacher tools

Work it out

Not feeling ready yet? These can help:

I.1Find derivatives using implicit differentiation

H.14Find derivatives using the chain rule I

Excellent! You got that right!

Jumping to level 1 of 1

Excellent!