# Thread: applying chain rule to natural log

1. ## applying chain rule to natural log

Hi, I'm very confused about these:
1. y=2^ln x
2. y=ln2^x

for #1, wouldn't it be:
y=2^ln x
y'=ln x (2^ln x) (ln 1/x) ?

but the answer in the book says it's y'=ln x (2^ln x) (1/x).

and I have absolutely no idea how to begin #2.

edit:
sorry, but these too:
3. f(x) = Ae^-Bx
book says f'(x) = Ae^-Bx (-B). I don't understand where would the -Bx go
4. f(x) = L/(1+Ae^-Bx)

thank you so much in advance!

2. Originally Posted by colloquial
Hi, I'm very confused about these:
1. y=2^ln x
2. y=ln2^x

for #1, wouldn't it be:
y=2^ln x
y'=ln x (2^ln x) (ln 1/x) ?

but the answer in the book says it's y'=ln x (2^ln x) (1/x).

and I have absolutely no idea how to begin #2.

thank you so much in advance!
For number 1 use the property that $\frac{d}{du}a^u = a^u\,ln(u)$

a = 2, u = ln(x)

$\frac{d}{dx}(2^{ln(x)}) = 2^{ln(x)}\,ln(ln(x)).$

Perhaps 1/ln(x) is obtained from the chain rule but I don't see it.

For number 2 use the property of logs that says $ln(b^a) = a\,ln(b)$

$y = ln(2^x) = x\,ln(2)$

This should be easy to differentiate. Remember ln(2) is a constant

3. Originally Posted by e^(i*pi)
For number 1 use the property that $\frac{d}{du}a^u = a^u\,ln(u)$
I would suggest you NOT use that property as it isn't true! The correct property is $\frac{a^u}{du}= ln(a) a^u$. Write $a^u= e^{ln(a^u)}= e^{uln(a)}$ and use the chain rule: the derivative is $e^{u ln(a)}$ times the derivative of u ln(a) with respect to u which is ln(a), not ln(u).

a = 2, u = ln(x)

$\frac{d}{dx}(2^{ln(x)}) = 2^{ln(x)}\,ln(ln(x)).$

Perhaps 1/ln(x) is obtained from the chain rule but I don't see it.
The derivative of $2^{ln(x)}$ is, by the chain rule, again, $ln(2) 2^{ln(x)}$ times the derivative of ln(x) which is 1/x. The derivative of $2^{ln(x)}$ is $ln(2)\frac{2^{ln(x)}}{x}$ so your book (believe it or not!) is correct.

For number 2 use the property of logs that says $ln(b^a) = a\,ln(b)$

$y = ln(2^x) = x\,ln(2)$

This should be easy to differentiate. Remember ln(2) is a constant
Exactly right!