Y= sin (2x^2 - 3e^x) Find dy/dx?

Apr 28, 2018

$\frac{\mathrm{dy}}{\mathrm{dx}} = \cos \left(2 {x}^{2} - 3 {e}^{x}\right) \cdot \left(4 x - 3 {e}^{x}\right)$

Explanation:

$y = \sin \left(2 {x}^{2} - 3 {e}^{x}\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \cos \left(2 {x}^{2} - 3 {e}^{x}\right) \cdot \left(4 x - 3 {e}^{x}\right)$

Apr 28, 2018

$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(4 x - 3 {e}^{x}\right) \cos \left(2 {x}^{2} - 3 {e}^{x}\right)$

Explanation:

Here,

$y = \sin \left(2 {x}^{2} - 3 {e}^{x}\right)$

Let, $u = 2 {x}^{2} - 3 {e}^{x} \implies \frac{\mathrm{du}}{\mathrm{dx}} = 4 x - 3 {e}^{x}$

$\mathmr{and} y = \sin u \implies \frac{\mathrm{dy}}{\mathrm{du}} = \cos u$

$\text{Using "color(blue)"Chain Rule}$

color(blue)((dy)/(dx)=(dy)/(du)*(du)/(dx).

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = \cos u \times \left(4 x - 3 {e}^{x}\right) , w h e r e , u = 2 {x}^{2} - 3 {e}^{x}$

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = \cos \left(2 {x}^{2} - 3 {e}^{x}\right) \times \left(4 x - 3 {e}^{x}\right)$

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = \left(4 x - 3 {e}^{x}\right) \cos \left(2 {x}^{2} - 3 {e}^{x}\right)$