Question

If , y = Sin ( π . Cos² x ) . Then find dy/dx

Answer :- 2πx ​

Answer 1

Answer:

0 I the write answer of the Question

Answer 2

$\bold \red{\frac{d}{dx} \left(y = \sin{\left(\pi \cos^{2}{\left(x \right)} \right)}\right)}$

SOLUTION

Differentiate separately both sides of the equation (treat y as a function of x):

$\bold \red{\frac{d}{dx} \left(y{\left(x \right)}\right) = \frac{d}{dx} \left(\sin{\left(\pi \cos^{2}{\left(x \right)} \right)}\right)}$

Differentiate the RHS of the equation.

$\bold \red{The \: function \: \sin{\left(\pi \cos^{2}{\left(x \right)} \right)}is \: the \: composition \: f{\left(g{\left(x \right)} \right)} \: of \: two \: functions \: f{\left(u \right)} = \sin{\left(u \right)} \: and \: g{\left(x \right)} = \pi \cos^{2}{\left(x \right)}}$

Apply the chain rule

$\bold \red{ \small{\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)}}$

$\bold{\color{red}{\left(\frac{d}{dx} \left(\sin{\left(\pi \cos^{2}{\left(x \right)} \right)}\right)\right)} = \color{red}{\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(\pi \cos^{2}{\left(x \right)}\right)\right)} \: </p><p>The \: derivative \: of \: the \: sine \: is \: \frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}}$

$\bold{\color{red}{\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\pi \cos^{2}{\left(x \right)}\right) = \color{red}{\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(\pi \cos^{2}{\left(x \right)}\right)}$

Return to the old variable:

$\bold \red{\cos{\left(\color{red}{\left(u\right)} \right)} \frac{d}{dx} \left(\pi \cos^{2}{\left(x \right)}\right) = \cos{\left(\color{red}{\left(\pi \cos^{2}{\left(x \right)}\right)} \right)} \frac{d}{dx} \left(\pi \cos^{2}{\left(x \right)}\right)</p><p>Apply \: the \: constant \: multiple \: rule \: \frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right) with \: c = \pi \: and \: f{\left(x \right)} = \cos^{2}{\left(x \right)}}$

$\bold \red{\cos{\left(\pi \cos^{2}{\left(x \right)} \right)} \color{red}{\left(\frac{d}{dx} \left(\pi \cos^{2}{\left(x \right)}\right)\right)} = \cos{\left(\pi \cos^{2}{\left(x \right)} \right)} \color{red}{\left(\pi \frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right)\right)} \: </p><p>The \: function \cos^{2}{\left(x \right)} \: is \: the \: composition \: f{\left(g{\left(x \right)} \right)} \: of \: two \: functions \: f{\left(u \right)} = u^{2} and g{\left(x \right)} = \cos{\left(x \right)}}$

Apply the chain rule

$\small\bold\red{\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)}$

$\bold \red{\pi \cos{\left(\pi \cos^{2}{\left(x \right)} \right)} \color{red}{\left(\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right)\right)} = \pi \cos{\left(\pi \cos^{2}{\left(x \right)} \right)} \color{red}{\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}</p><p>Apply \: the \: power \: rule \: \frac{d}{du} \left(u^{n}\right) = n u^{n - 1} \: with \: n = 2}$

$\bold \red{\pi \cos{\left(\pi \cos^{2}{\left(x \right)} \right)} \color{red}{\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) = \pi \cos{\left(\pi \cos^{2}{\left(x \right)} \right)} \color{red}{\left(2 u\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)}$

Return to the old variable:

$\bold \red{2 \pi \cos{\left(\pi \cos^{2}{\left(x \right)} \right)} \color{red}{\left(u\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) = 2 \pi \cos{\left(\pi \cos^{2}{\left(x \right)} \right)} \color{red}{\left(\cos{\left(x \right)}\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)</p><p>The \: derivative \: of \: the \: cosine \: is \: \frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}}$

$\bold \red{2 \pi \cos{\left(x \right)} \cos{\left(\pi \cos^{2}{\left(x \right)} \right)} \color{red}{\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = 2 \pi \cos{\left(x \right)} \cos{\left(\pi \cos^{2}{\left(x \right)} \right)} \color{red}{\left(- \sin{\left(x \right)}\right)}</p><p>Thus, \frac{d}{dx} \left(\sin{\left(\pi \cos^{2}{\left(x \right)} \right)}\right) = - 2 \pi \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\pi \cos^{2}{\left(x \right)} \right)}}$

Therefore,

$\bold \red{{\frac{dy}{dx} = - 2 \pi \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\pi \cos^{2}{\left(x \right)} \right)}}}$

Answer

$\bold \red{{\frac{dy}{dx} = - 2 \pi \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\pi \cos^{2}{\left(x \right)} \right)}}}$