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I was trying to understand the proof of

(d/dx) b^x = (ln b)*(b^x)

it says:

b= e^(ln b)

so, b^x= e^((ln b)*x))

So now we use the chain rule:

(d/dx) b^x = (d/dx) e^((ln b)*x))

I understand everything so far, but not the next step.

It says then that

(d/dx) e^((ln b)*x))=(ln b)*e^(ln b)*x)

how did they get this, i am confused about the bold part. I am forgetting something about taking the derivative of the natural number 'e'?

I know that d/dx e^x = x

so how does the ln b come behind e^x above?

what does (d/dx) e^f(x) equal to?

is (d/dx) e^f(x) = f '(x)* e^f(x) ?

if so then

1 + ln b

should come behind that because

d/dx (ln b)*x is equal to 1 + ln b

Please help me

thank you

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# Homework Help: Derivatives of natural numbers

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