How to derivate f(x)^g(x)?

Question

Do you know how to derivate f(x)^g(x)?
I mean, for example, to derivate f(x)*g(x) you do: f'(x)*g(x) + f(x)*g'(x).

Answer 1

It can be done with chain-rule and product rule.

rewrite f^g as follows:

y = f^g = e^(ln(f)*g)



Now use chain-rule and product rule.

y' = (f'g/f + ln(f)*g')*e^(ln(f)*g) = (f'g/f + ln(f)*g') * f^g

Answer 2

To do f^g you'll need a technique known as logarithmic differentiation. Simply equate the expression to y

y = f^g



Now take the ln of both sides

ln(y) = ln(f^g) = g ln(f)



Now differentiate implicitly taking f, g, and y as functions in x

y' / y = g' ln(f) + g(f' / f)

y' = y(g' ln(f) + g(f' / f))



Now substitute back in y

y' = (f^g)(g' ln(f) + g(f' / f))

Answer 3

To do f^g you'll need a technique known as logarithmic differentiation. Simply equate the expression to y

y = f^g



Now take the ln of both sides

ln(y) = ln(f^g) = g ln(f)



Now differentiate implicitly taking f, g, and y as functions in x

y' / y = g' ln(f) + g(f' / f)

y' = y(g' ln(f) + g(f' / f))



Now substitute back in y

y' = (f^g)(g' ln(f) + g(f' / f))

Answer 4

g(x) * (f(x)^(g(x)-1)) * f'(x) + (ln f(x)) * f(x)^g(x) * g'(x)