How do you apply the power rule when E is raised by an power to a power?

f[x] = e^(x^2)
f'[x] = ???

Question

How do you apply the power rule when  E is raised by an power to a power?

f[x] = e^(x^2)
f'[x] = ???

thesmartone · Answer

@Michele_Laino QH help :)

amistre64 · Answer

derivative or integration?

mathstudent55 · Answer

You don;t use the power rule here.
Here you need the derivative of e^u, where u is a function in x.

amistre64 · Answer

i see, derivative, chain and then power

anonymous · Answer

trying to get to derivative.. yes

mathstudent55 · Answer

$\dfrac{d}{dx} e^u = e^u \dfrac{d}{dx}u$

michele_laino · Answer

here we have to apply the derivation of composed functions

mathstudent55 · Answer

Oh, I see. The power rule does come in to differentiate the exponent, x^2.

anonymous · Answer

so its like 2 functions?

michele_laino · Answer

noe we can write your function as follows:
$$f\left( x \right) = {e^{g\left( x \right)}},\quad g\left( x \right) = {x^2}$$

michele_laino · Answer

so the requested first derivative is given by the subsequent formula:
$$f'\left( x \right) = \frac{{df}}{{dg}} \times \frac{{dg}}{{dx}} = {e^{g\left( x \right)}} \times 2x = 2x{e^{{x^2}}}$$

anonymous · Answer

wicked  .. thank you
now I just need to get my head around that.. :)
... first day solving derivative problems using rules. .. :)

michele_laino · Answer

thank you! :)

anonymous · Answer

makes sense though :)

michele_laino · Answer

need more help?

anonymous · Answer

all good thnx

michele_laino · Answer

thanks!! :)