If f(x) = e^(5x), is f´(x) = e^(5x)?

$$f(x)=e^{5x}\\\\
f'(x)=5e^{5x}$$

 

You can memorize this outcome but if you understand it that would be even better.

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you know that if  $$y=e^x$$  then  $$\frac{dy}{dx}=e^x$$

Now

consider $$y=e^{5x}$$

Let u=5x  then   $$y=e^u\quad \mbox{and}\quad \frac{du}{dx}=5$$

 

$$y=e^u$$

Now use the chain rule:

$$\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}=e^u\times 5 = 5e^{5x}$$

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