How do you differentiate #f(x)=e^(4^(1/(x^2)))# using the chain rule?

1 Answer
Jim H
Nov 11, 2015

I read this as #e# to the power of (#4# raised to the #1/x^2# power or #4^(x^-2)#). For the derivative of #f(x) = e^(4^(1/x^2)) = e^(4^(x^-2))#, see below.

Explanation:

The derivative of #e^u# with respect to #x# is #e^u (du)/dx#.

So the first step gives us:

#f'(x) = e^(4^(x^-2)) d/dx(4^(x^-2))#

The derivative with respect to #x# of #4^u# is #4^u ln4 (du)/dx#

So our second step gets us:

#f'(x) = e^(4^(x^-2)) (4^(x^-2) ln4 d/dx(x^-2))#

And the third step gets us:

#f'(x) = e^(4^(x^-2)) (4^(x^-2) ln4 (-2x^-3))#

We can clean this up a little bit:

#f'(x) = -2/x^3 4^(x^-2)e^(4^(x^-2)) ln4#

# = -2/x^3 4^(1/x^2)e^(4^(1/x^2)) ln4#

Other expressions are also possible.

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