You are Here: Home >< Maths

# Is the derivative of a^f(x) = f'(x) (a^x)(lna) ? Watch

1. And the integral of a^f(x) = (a^f(x))/(f'(x) x lna) ?
2. The integral you said isn't correct. You can't apply the reverse chain rule in that way as you get a product, so when you differentiate it you use the product rule(or quotient) and you will see that it's not right.
3. hope this helps dy/dy(2^x)dx =2^xln(2) so the rule is a^x= a^xln(a)

Proof........ dy/dx (a^x) dx....let y=a^x so ln(y)=ln(a^x) so ln(y)= xln(a) ... implicit differentiation so 1/y dy/dx =ln(a) so bringing the y up .... dy/dx = yln(a) ans as y=a^x dy/dx = a^xln(a)
4. (Original post by chhhhelsie)
And the integral of a^f(x) = (a^f(x))/(f'(x) x lna) ?
Your derivative formula is correct, but the integral one is not, and in fact it is one of the most common misconceptions among students - specifically, the "reverse chain rule", which many students believe is true in general, in fact only applies when f(x) is linear (of the form ax+b). Thus for example the integral of e^(x^2), letting f(x)=x^2 and a=e in other words, is not e^(x^2)/2x - in fact, this function e^(x^2) is impossible to integrate, although the proof of this result is beyond A-Level.
5. Integral (a^xln(a)) dx = a^x as its the thing you differentiated so go back

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: June 20, 2016
Today on TSR

### What is the latest you've left an assignment

And actually passed?

### Simply having a wonderful Christmas time...

Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.