what is the dervative of y= lnx^3?

Answer 1

Is this ln(x^3) or (ln x)^3?

If it's the first, the answer is y' = 3/x.

If it's the second, the answer is y' = 3 (ln x)^2 / x.

Cheers.

Answer 2

There are two ways you can approach this. You can either use the log rule and then take the derivative, or take the derivative now. I'll show you both methods.

Right now, I'll solve it as is.

y = ln(x^3). Then, since the derivative of ln(x) is 1/x, but the inside is NOT x, by the chain rule, we have to multiply by the derivative of the inside (x^3).

y' = [1/(x^3)] [3x^2]
y' = [3x^2]/[x^3] = 3/x

Method 2: Recognize that y = ln(x^3) is the same as y = 3ln(x); that is, by the log rule that

log[base b](a^c) = c * log[base b](a)

So let's differentiate.

y = 3ln(x)
y' = 3(1/x) = 3/x, which is the EXACT same answer as above; the only difference is, no chain rule was involved.

Answer 3

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Answer 4

the derivative of lnx =1/x

1/x^3 (x^3)' =
1/x^3 (3x^2)=
3x^2/x^3=
3/x

Answer 5

Well, it depends on if you mean ln(x^3) or (ln(x))^3.

For ln(x^3), dy/dx = 3/x

For (ln(x))^3, dy/dx = (3 ln(x)^2)/x

Okay, first one is pretty simple, pull out the 3 with a log rule and then just derive.

The second one is a composite deriv.

Answer 6

use the chain rule


3x^2 /(x^3) = 3/x

Answer 7

1/x^3 *3x^2
=3/x

Answer 8

y'=(1/x^3)*3x^2 //chain rule, power rule
y'=3x^2/x^3 //Simplify
y'=3/x

Answer 9

y= ln(x^3)
y' = (3x^2)/x^3
y' = 3/x