# Thread: simplify log derivative/distribution

1. ## simplify log derivative/distribution

I have been having trouble trying to follow along an example of, finding the derivative of $\displaystyle (11x)^{ln11x}$ .

Using the property $\displaystyle A^r = rlnA$

$\displaystyle ln(f(x))= (ln11x) (ln11x)$

Now differentiate the left side, then the right side....

Left Side: $\displaystyle \frac{d}{dx}[ln(f(x))]= \frac{1}{f(x)}f'(x)$

Right side:$\displaystyle \frac{d}{x}[(ln11x) (ln11x)] = \frac{2ln11x}{x}$

$\displaystyle \frac{1}{f(x)}f'(x) = \frac{2ln11x}{x}$

multiply both sides by f(x) , to get f'(x) by itself. $\displaystyle f'(x) = f(x)* \frac{2ln11x}{x}$

Now replcace f(x) for $\displaystyle (11x)^{ln11x}$

$\displaystyle f'(x) = (11x)^{ln11x} * \frac{2ln11x}{x}$

I getting confused at this part...... how does it get to here

$\displaystyle = 2* 11^{ln11x}x^{ln11x-1}ln11x$

2. ## Re: simplify log derivative/distribution

$\displaystyle (11x)^{ln11x} = 11^{ln11x}x^{ln11x}$

So

$\displaystyle \frac{(11x)^{ln11x}}x= \frac{11^{ln11x}x^{ln11x}}x= 11^{ln11x}x^{ln11x-1}$

Now multiply both sides by $\displaystyle 2ln11x$ and you are there.