How do you differentiate #f(x)=e^tan(x) # using the chain rule?

1 Answer
Ken C.
Apr 25, 2016

Multiply the derivative of #e^tanx# by the derivative of #tanx# to get #f'(x)=e^(tanx)sec^2x#.

Explanation:

Differentiating this will require use of the chain rule, which, put plainly, states that the derivative of a composite function (like #e^tanx#) is equal to the derivative of the "inside function" (in this case #tanx#) multiplied by the derivative of the whole function (#e^tanx#).

In math terms, we say the derivative of the composite function #f(g(x))# is #f'(g(x))*g'(x)#.

So, the derivative of #e^tanx# will be the derivative of #e^tanx#, which is just #e^tanx# (the derivative of #e# to the anything is #e# to the anything) times the derivative of #tanx#, which is #sec^2x#. That is to say:
#d/dxe^tanx=e^tanx*(tanx)'=e^tanxsec^2x#

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