Question

composite function of derivative find y= e^2x•tan3x​

Answer 1

${\underline{\sf{Question}}}$

Function:

$\sf \: y = e {}^{2x} \times \tan3x$

${\underline{\sf{To\:Find}}}$

Derivative of given function

${\underline{\sf{Theory}}}$

${\red{\boxed{\large{\bold{Composite\: Function (Chain\:Rule)}}}}}$

Let y=f(t) ,t = g(u) and u =m(x) ,then

$\sf\:\dfrac{dy}{dx} = \dfrac{dy}{dt} \times \dfrac{dt}{du} \times \dfrac{du}{dx}$

${\underline{\sf{Solution}}}$

$\sf \: y = {e}^{2x} \times \tan3x$

Differentiate with respect to x , by chain rule .

$\sf \implies \dfrac{dy}{dx} = [e {}^{2x} \times \dfrac{d ( \tan3x)}{d(3x)} \times \dfrac{d(3x)}{dx} + \tan3x \times \dfrac{d( {e}^{2x} )}{d(2x)} \times \dfrac{d(2x)}{dx} ]$

$\sf \implies \dfrac{dy}{dx} = [ {e}^{2x} \times3 \times \sec {}^{2}3x + \tan3x \times 2 \times {e}^{2x} ]$

$\sf \implies \dfrac{dy}{dx} = [ 3{e}^{2x} \sec {}^{2}3x + 2\tan3x {e}^{2x} ]$

It is the required solution!

$\rule{200}2$

${\underline{\sf{Formula's}}}$

$1)\sf\:\frac{d(x {}^{n} )}{dx} = nx {}^{n - 1}$

$2)\sf\:\frac{d(constant)}{dx} = 0$

$3) \sf\dfrac{d( \tan \: x)}{dx} = \sec{}^{2}\: x$

$\sf4) \dfrac{d(e {}^{x}) }{dx} = {e}^{x}$