Question

Differentiating a Logarithmic Function In Exercise, find the derivative of the function.

Answer 1

Answer:

$\frac{d}{dx} (ln(e^{x^2}) ) = \frac{1}{e^{x^2}} 2x e^{x^2} = 2x$

Step-by-step explanation:

For this case we want to find the derivate of this function:

$y = ln(e^{x^2}) )$

And in order to find the derivate we need to apply the chain rule given by:

$\frac{df(u)}{dx} =\frac{df}{du} \frac{du}{dx}$

And on this case $f = ln(u), u = e^{x^2}$

And we can find the partial derivates like this:

$\frac{d}{du} (ln(u)) =\frac{1}{u}$

$\frac{d}{dx}(e^{x^2})= e^{x^2} (2x)$

And if we replace we got:

$\frac{d}{dx} (ln(e^{x^2}) ) = \frac{1}{u} 2x e^{x^2}$

And if we replace$u = e^{x^2}$ we got:

$\frac{d}{dx} (ln(e^{x^2}) ) = \frac{1}{e^{x^2}} 2x e^{x^2} = 2x$

And that would be our final answer on this case