How do you differentiate y=e^(2x^3)?

Oct 5, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = 6 {x}^{2} {e}^{2 {x}^{3}}$

Explanation:

We use Chain Rule here.

In order to differentiate a function of a function, say $y , = f \left(g \left(x\right)\right)$, where we have to find $\frac{\mathrm{dy}}{\mathrm{dx}}$, we need to do substitute $u = g \left(x\right)$, which gives us $y = f \left(u\right)$.

The Chain Rule states that $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \times \frac{\mathrm{du}}{\mathrm{dx}}$.

In fact if we have something like $y = f \left(g \left(h \left(x\right)\right)\right)$, we can have $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{df}} \times \frac{\mathrm{df}}{\mathrm{dg}} \times \frac{\mathrm{dg}}{\mathrm{dh}}$

Here we have $y = {e}^{2 {x}^{3}}$

Hence $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{d \left({e}^{2 {x}^{3}}\right)}{d \left(2 {x}^{3}\right)} \times \frac{d \left(2 {x}^{3}\right)}{\mathrm{dx}}$

= ${e}^{2 {x}^{3}} \times 2 \times 3 {x}^{2}$

= $6 {x}^{2} {e}^{2 {x}^{3}}$