d/dx(e^5 ln(tan 5x))?

1 Answer
turksvids
Dec 27, 2017

#25tan^4(5x)#

Explanation:

First rewrite the original function: #e^(5ln(tan(5x))) = e^(ln(tan^5(5x)))# by the property of natural logs that coefficients become exponents.

Now look at the function #e^(ln(tan^5(5x)))# and realize it can be simplified because #e^x# and #ln(x)# are inverses, so #e^ln(u) = u#. So really our function is just #tan^5(5x)#.

Now we can find the derivative using the chain rule:

#d/dx(tan^5(5x)) = 5*tan^4(5x)*5 = 25tan^4(5x)#. The first 5 is from derivative of #u^5# and the second 5 is from derivative of #5x#.

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