Simplification of the derivative

Question

I have this equation $y=x^{5x^3}$ by doing a log transformation we get, $log (y) = 5x^3 log (x)$ upon doing a differentiation w.r.t $(x)$, we get

$$\frac{1}{y}\frac{dy}{dx} = 5x^3.\frac{1}{x} + log(x) . 15x^2 =>\frac{dy}{dx} = x^{5x^3}(5x^3.\frac{1}{x} + log(x) . 15x^2)$$

upon simplification we get $$\frac{dy}{dx}=5x^{5x^3+2}(1+3 log (x))$$

but the result says $$\frac{dy}{dx}=5x^{5x^3+2}.log (e^ {x^3})$$

Can you please tell me how do I simplify the whole equation to get the result

Thanks,
Kamal.

score 2 · Accepted Answer · answered Mar 05 '15 at 09:18

2

Your result is perfectly correct and the last is totally wrong. It could be $\log(e x^3)$ obtained from your since $$1+3\log(x)=\log(e)+\log(x^3)=\log(e x^3)$$ but $\log(e^{x^3})$ is perfectly wrong.

May be another typo in the book.

answered Mar 05 '15 at 09:18

Claude Leibovici

260,315

Is log same as ln? ln(e) =1 but log(e), so just want to know how is this done? – Kamal Mar 05 '15 at 11:14
1

Yes. For most people (I think) $\log$ stands for $\ln$. For any other base $a$, inclusing $10$, most people use to write $\log_a$ – Claude Leibovici Mar 05 '15 at 11:23
You are very welcome ! – Claude Leibovici Mar 05 '15 at 11:35

Simplification of the derivative

1 Answers1