Differentiation of the log, cubed
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We wish to find
.
First let’s discuss the function
f
.
Is the final operation to do a “ln” function or a cubing function?
The final operation is a cubing.
Does
f
require taking the natural log three times?
No.
What is to be cubed?
Just the expression in the square brackets.
And that expression is …..
ln(4
x
-5)
To summarize, the function
f
involves taking the ln of (4
x
-5),
and then cubing it.
What mathematical term from algebra describes this application of a function, like cubing, to the result of another function, like taking the natural log?
Composition of functions
.
To take the derivative of
f
, what rules of differentiation
do we need beside the Rule for ln:
?
We need to use the
Power Rule
and the
Chain Rule
.
What do we differentiate first – the natural log or the power
We start on the “outer” function first.
Which is that here?
The power is the last operation, or the outer function.
Set that up.
Now take the derivative of the “ln”.
We get
Do the last indicated derivative.
Simplify.
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General Contents
Detailed Contents
Index