Hyperbolic Differentiation: How Do We Differentiate Functions with Exponents?

[DiamondV]

Homework Statement

Differentiate

Homework Equations

Chain Rule: dg/dx = du/dx . dv/du . dg/dv

The Attempt at a Solution

My answer(wrong):

Correct answer provided to us(not mine):

I understand the correct solution that was provided to us, but what I don't understand is why my method isn't correct? Also can you check in particular my dv/du . I suspect there's something wrong there.

 

[pasmith]

The derivative of $3^v = e^{v \log 3}$ is $(\log 3) e^{v \log 3} = (\log 3)3^v$, not $3v^{v-1}$ which appears to be what you have.

If you set $v = u\log x$ then you want to calculate $$\frac{dv}{du} = \log x + \frac ux \frac{dx}{du}$$ and the $\frac{dx}{du}$ cancels with $\frac{du}{dx}$ when you apply the chain rule.

Differentiating functions in exponents is straightforward: By the chain rule, with $a > 0$, $$\frac{d}{dx} a^{f(x)} = (\log a)a^{f(x)} f'(x).$$