Question

Find the derivative of 5^(x^3+x)

I know the answer, it's:

5^(x^3+x)*ln(5)*(3x^2+ 1)

I just need some help on how to derive that. What rule is at work here? where did the ln come from?

Answer 1

I assume you know the chain rule, and that

d/dx e^u = e^u du/dx

Now, for some other base, such as 5, recall that

5 = e^ln5

So, 5^x = e^(ln5 x)
d/dx 5^x = ln5 e^(ln5 x) = ln5 5^x

So we arrive at the fact that

d/dx a^u = ln(a) a^u du/dx

and your result follows.