A couple derivative problems: I've worked through them but..

[mmc5311]

..are my answers correct? I'm not very confident in my calculus skills, haha.

1. y = [ln(2x-1)]^2

What I did:

a.) moved the exponent to the front. 2 [ln(2x-1)]
b.) took the derivative of ln(2x-1). 1/(2x-1)
c.) took 2 * (1/(2x-1)) and got 2/(2x-1) as the derivative. Is it correct? If not what did I do wrong?


2. y = x^5e^(-3lnx)

What I did:
a.) moved the -3 from the front of lnx and tacked it on as the exponent. x^5e^(lnx^-3)
b.) let the e^ln cancel each other out, and was left with x^5*x^-3
c.) I got x^2, and I found the derivative of that: 2x. Correct or incorrect?


Thanks again =)

 

[DrSteve]

Re: A couple derivative problems: I've worked through them b

Looks like you have the right idea in the first one. The final answer is 2[ln(2x-1)]2/(2x-1) = 4ln(2x-1)/(2x-1)

The second one is correct.

 

[soroban]

Re: A couple derivative problems: I've worked through them b

Hello, mmc5311!

\(\displaystyle 1.\;y \:=\:[\ln(2x-1)]^2\)

\(\displaystyle \text{What I did:}\)

\(\displaystyle \text{a) moved the exponent to the front. }\:y \:=\:2[\ln(2x-1)]\) . No

\(\displaystyle \text{You could do that if we had: }\:y \:=\:\ln(2x-1)^2 \quad\hdots\quad \text{but we don't.}\)


\(\displaystyle \text{We have: }\:y \:=\:[f(x)]^2\)

. . \(\displaystyle \text{The derivative is: }\:y' \;=\;2\cdot [f(x)]\cdot f'(x)\)


\(\displaystyle \text{The answer is: }\;y' \;=\;2[\ln(2x-1)]\cdot \frac{1}{2x-1}\cdot 2 \;=\;\frac{4\ln(2x-1)}{2x-1}\)

\(\displaystyle 2.\;y \:=\:x^5e^{-3\ln x}\)

What I did:

. . a) moved the \(\displaystyle -3\) from the front of \(\displaystyle \ln x\) and tacked it on as the exponent: .\(\displaystyle y \:=\:x^5e^{\ln x^{-3}}\)

. . b) let the \(\displaystyle e^{\ln}\) cancel each other out, and was left with \(\displaystyle y \:=\:x^5\cdot x^{-3}\)

. . c) I got \(\displaystyle y \:=\:x^2\), and I found the derivative of that: \(\displaystyle y' \:=\: 2x\).

Correct or incorrect?

Correct! . . . Good work!

 

[mmc5311]

Re: A couple derivative problems: I've worked through them b

Great. Thanks for the help