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## Homework Statement

1. If f(x), g(x) and h(x) are real functions of x, show that

when [tex]h(x)=[f(x)]^{g(x)}[/tex]

then [tex]h'(x)=[f(x)]^{g(x)}(g'(x)\ln[f(x)]+g(x)\frac{f'(x)}{f(x)})[/tex]

2. [tex]\frac{A}{x}+\frac{B}{P-x}=\frac{1}{x(P-x)}[/tex] where x is a variable, and P is a constant. Find A and B in terms of P.

## Homework Equations

## The Attempt at a Solution

1. I start by doing what I usually do, like with [tex]x^{x^2}[/tex]:

[tex][f(x)]^{g(x)-1}.g(x).g'(x)[/tex]

Looking at the derivative, I see

[tex]\int{g'(x)\ln[f(x)]+g(x)\frac{f'(x)}{f(x)}}=g(x)\ln[f(x)][/tex]

Which looks nothing like what I got :(

2. Getting a common denominator and canceling:

[tex]A(P-x)+Bx=1[/tex]

Then by inspection,

[tex]A=\frac{P}{P^2}[/tex]

[tex]B=\frac{1}{P^2}[/tex]

It was a fluke that I got that :/. So I'm wondering how to prove it arithmetically, or just some general method of solving these kinds of problems for when I come across them again.