Question

Differentiate w.r.t x

Answer 1

f'(x) = -1 .

Explanation is above on the pic.

Hope it helps to you.

Answer 2

Question :–

$\\ { \bold{Differentiate \: \: w.r.t \: \: x \: \: : \: { \cos}^{ - 1} \left( \dfrac{ \sin(x) + \cos(x) }{ \sqrt{2} } \right)}}$

ANSWER :–

• Let the function be –

$\\ \implies { \bold{y \: = \: { \cos}^{ - 1} \left( \dfrac{ \sin(x) + \cos(x) }{ \sqrt{2} } \right)}}$

$\\ \implies { \bold{y \: = \: { \cos}^{ - 1} \left( \dfrac{ \sin(x) }{ \sqrt{2} } + \frac{ \cos(x) }{ \sqrt{2} } \right)}}$

• We know that –

$\\ \longrightarrow \: \: \large { \green { \boxed { \bold{ \sin( \dfrac{\pi}{4} ) = \cos( \dfrac{\pi}{4} ) = \frac{1}{ \sqrt{2} } }}}}$

• So that –

$\\ \implies { \bold{y \: = \: { \cos}^{ - 1} \left[ \sin( \dfrac{\pi}{4} ) { \sin(x) } + \cos( \dfrac{\pi}{4} ) { \cos(x) } \right]}}$

• Now using identity –

$\\ \large \star \: \: { \green{ \boxed{ \bold{ \cos(a - b) = \sin( a) { \sin(b) } + \cos( a ) { \cos(b) } }}}}$

$\\ \implies{ \bold{y \: = \: { \cos}^{ - 1} \left[ \cos( \dfrac{\pi}{4} - x ) \right]}}$

• According to another condition –

$\\ \implies{ \bold{ - \dfrac{\pi}{4} < x < \dfrac{\pi}{4} }}$

$\\ \implies{ \bold{ \dfrac{\pi}{4} + ( - \dfrac{\pi}{4}) < (\dfrac{\pi}{4} - x) < \dfrac{\pi}{4} + \dfrac{\pi}{4} }}$

$\\ \implies{ \bold{ 0 < (\dfrac{\pi}{4} - x) < \dfrac{\pi}{2} }}$

$\\ \implies{ \bold{ ( \dfrac{\pi}{4} - x) \in (0, \frac{\pi}{2}) }}$

$\\ \: \because \: { \bold{ ( \dfrac{\pi}{4} - x) \: \: belongs \: \: in \: \: principal \: \: value . }}$

$\\ \: \therefore \: { \bold{ we \: \: should \: \: write \: \: this \: \: as -}}$

$\\ \implies{ \bold{y \: = ( \dfrac{\pi}{4} - x ) }}$

• Now Differentiate with respect to 'x' –

$\\ \large \implies{ \red{ \boxed{ \bold{ \dfrac{dy}{dx} \: = - 1}}}}$

$\\ \rule{220}{2}$