Derivative of e^2x?

Question

What is the derivative of e when it's raised to something other than just e^x

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Answer 1 · 2017-10-12T09:10:53

#d/(dx)(e^(2x))=2e^(2x)#

in general for

#y=e^(f(x))#

we just need to use the chain rule

#(dy)/(dx)=(dy)/(du)(du)/(dx)#

#u=f(x)=>(du)/(dx)=f'(x)#

#y=e^u=>(dy)/(du)=e^u#

#:.(dy)/(dx)=e^uxxf'(x)=f'(x)e^(f(x))#

#:.color(blue)(d/(dx)(e^(f(x)))=f'(x)e^((f(x))#

it would be a good idea to memorise the above result.

with #y=e^(2x)# we case differentiate immediately

#d/(dx)(e^(2x))=2e^(2x)#