find f'(x) if f(x)=5e^x^2

Question

find f'(x) if f(x)=a^x

Answer 1

f = 5e^(x^2))
use chain rule:

f = 5e^u
f' = 5e^u u'
f' = 5e^(x^2) * 2x = 10x e^(x^2)

f = a^x
recall that a = e^(ln a)
f = e^(lna * x)
f' = lna * e^(lna * x) = a^x lna

Answer 2

Just something you may find interesting:

y = u^v
lny = v lnu
1/y y' = lnu v' + v/u u'
y' = u^v lnu v' + vu^(v-1) u'

Note that

if v is a constant n, v' = 0 and y' = nu^(n-1) u' is just the power rule

if u is a constant a, u' = 0 and y' = a^v lna v' as in your problem.

cool, eh?

Answer 3

so a^x would be undefined?

Answer 4

no. a is just a number, like e.
Is e^x undefined?

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