Yay calculus.
First (and you probably know this), f'(x) is shorthand for df/dx
so if f'(x) = x[sup]3[/sup]
then f(x) = (1/4)(x[sup]4[/sup]) (anti-derivative of x[sup]3[/sup])
So f(u) = f(e[sup]x[/sup])
= (1/4)((e[sup]x[/sup])[sup]4[/sup])
= (1/4)(e[sup]4x[/sup])
d/dx((1/4)(e[sup]4x[/sup])) = (1/4) d/dx(e[sup]4x[/sup]) (Constant rule: If k is constant then d/dx(kx) = k(d/dx(x)))
Now apply the chain rule to the new derivative:
If h(x) = f(g(x))
then h'(x) = f'(g(x)) g'(x)
Letting
f(x) = e[sup]x[/sup]
and
g(x) = 4x
Now derive:
f(x) = e[sup]x[/sup] => f'(x)= e[sup]x[/sup] (e[sup]x[/sup] is its own derivative)
g(x) = 4x => g'(x) = (4)(1x[sup]0[/sup]) = 4
So h'(x) = (e[sup]4x[/sup])(4)
= 4e[sup]4x[/sup]
So back to d/dx((1/4)([sup]4x[/sup]))
= (1/4) d/dx(e[sup]4x[/sup])
= (1/4) (4e[sup]4x[/sup])
= (1/4) (4) (e[sup]4x[/sup])
= e[sup]4x[/sup]
I repeat:
yay calculus.
And I eagerly await a bulletin board that can handle MathML.
Also, I take the title "Pokemon Professor"
quite literally.