# Differentiating natural logs and exponential functions?

Hey all, I'm really having a hard time figuring out a couple of problems in which I have to differentiate:

1: $$\frac{e^3^x}{\ln x}$$

I just don't know how to put it together... I know that $$e^3^x$$ is $$3e^3^x$$, and I know that you can't different $$\ln x$$, so I dunno what to do from there...

And:

2: $$\ln(e^-^2^x + e^-^x)$$

Totally not sure what to do there... Any help would be appreciated, I've looked online for similar examples and couldn't really find anything relevant to help me. Thanks!

Hootenanny
Staff Emeritus
Gold Member
Let's take the first question first,
Hey all, I'm really having a hard time figuring out a couple of problems in which I have to differentiate:

1: $$\frac{e^3^x}{\ln x}$$

I just don't know how to put it together... I know that $$e^3^x$$ is $$3e^3^x$$
Correct.
and I know that you can't different $$\ln x$$, so I dunno what to do from there...
Erm... yes you can!

HINT: Let y = lnx, then x = ey. Now differentiate x = ey with respect to x.

Let's take the first question first,

Correct.

Erm... yes you can!

HINT: Let y = lnx, then x = ey. Now differentiate x = ey with respect to x.

Hrm, I'm feeling a bit retarded so bear with me, hehe. So, $$\frac {1}{x}$$?

Hootenanny
Staff Emeritus
Gold Member
Hrm, I'm feeling a bit retarded so bear with me, hehe. So, $$\frac {1}{x}$$?
Correct

So as far as putting it all together, can you suggest anything? Perhaps some sort of rule...?

I'm really retarded at math, ugh... Hehe. So I have $$\frac {3e^3^x}{1/x}$$ ... Pretty sure I'm lost on what to do from there. :(

Hootenanny
Staff Emeritus
Gold Member
I'm really retarded at math, ugh... Hehe. So I have $$\frac {3e^3^x}{1/x}$$ ... Pretty sure I'm lost on what to do from there. :(
No that is not correct.

$$\frac{d}{dx} \frac{f\left(x\right)}{g\left(x\right)} \neq \frac{f^\prime\left(x\right)}{g^\prime\left(x\right)}$$

What 'rules' of differentiation do you know?

Not sure of any rules by name... I'm going to have to research more examples, I'm so rusty in math that I am unsure...

Hootenanny
Staff Emeritus
Gold Member
Not sure of any rules by name... I'm going to have to research more examples, I'm so rusty in math that I am unsure...
Try searching for the quotient rule, or the product rule and chain rule.

I kinda give up on the first one, heh... Can someone tell me if I have #2 right?:

Answer: $$\frac {1 - e^-^2^x}{e^-^x + e^-^2^x}$$

Pengwuino
Gold Member
I kinda give up on the first one, heh... Can someone tell me if I have #2 right?:

Answer: $$\frac {1 - e^-^2^x}{e^-^x + e^-^2^x}$$

The quotient rule and product rule explains how to differentiate two functions that divide are divided by or multiplied by eachother. Just look that up and identify f(x) and g(x) (that is the standard notation at least).

For #2, it isn't correct. What did you do to come up with that answer?

Mirth, can you show us step by step how you differentiated that expression?

tiny-tim