# Question #bc6f3

Jun 5, 2015

We have to differentiate it by chain rule.

With practice, probably you won't need to introduce dummy variables. But to develop the initial grasp, its recommended that you should start by introducing variables at every step.

Now, the given function is in the form of $y = {e}^{3 {x}^{7}}$. You don't know how to solve it. However you are probably familiar with the concept of solving a function in the form of ${e}^{x}$. So, our 1st dummy variable $u = 3 {x}^{7}$

Thus, our modified function is now, $y = {e}^{u}$

Differentiating w.r.t. u, we get: $\frac{\mathrm{dy}}{\mathrm{du}} = {e}^{u}$

However we have to solve $\frac{\mathrm{dy}}{\mathrm{dx}}$

Now, $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{du}}{\mathrm{dx}} \cdot \frac{\mathrm{dy}}{\mathrm{du}}$

Differentiating $u$ w.r.t. $x$, we get: $\frac{\mathrm{du}}{\mathrm{dx}} = 3.7 . {x}^{6}$
$\implies \frac{\mathrm{du}}{\mathrm{dx}} = 21 {x}^{6}$

so, $\frac{\mathrm{dy}}{\mathrm{dx}} = 21 {x}^{6} \cdot {e}^{3 {x}^{7}}$