Question

Q1) Determine the derivative of cosx/(1+sinx).



☞ No COPPIED and IRRELEVANT ANSWERS ❌​

Answer 1

GIVEN :–

• Function => cosx / (1 + sinx)

TO FIND :–

• Derivative = ?

SOLUTION :–

• Let the function be –

$\\ \bf \implies y = \dfrac{ \cos(x) }{1 + \sin(x) }$

• Using identity –

$\\ \bf \implies \: \dfrac{d}{dx} \left( \dfrac{u}{v} \right) = \dfrac{v. \dfrac{du}{dx} - u.\dfrac{dv}{dx} }{ {v}^{2} }$

• Now Differentiate with respect to 'x' –

$\\ \bf \implies \dfrac{dy}{dx} = \dfrac{ \{1 + \sin(x) \} \dfrac{d( \cos(x)) }{dx} - \cos(x)\dfrac{d( 1 + \sin(x)) }{dx}}{ \{ 1 + \sin(x) \}^{2}}$

$\\\bf \implies \dfrac{dy}{dx} = \dfrac{ \{1 + \sin(x) \} \{ - \sin(x) \}- \cos(x) \{ \cos(x) \}}{ \{ 1 + \sin(x) \}^{2} }$

$\\\bf \implies \dfrac{dy}{dx} = \dfrac{ - \sin(x) - \sin^{2} (x) - \cos^{2} (x)}{ \{ 1 + \sin(x) \}^{2}}$

$\\ \bf \implies \dfrac{dy}{dx} = \dfrac{ - \sin(x) - \{ \sin^{2} (x) + \cos^{2} (x) \}}{ \{ 1 + \sin(x) \}^{2}}$

$\\ \bf \implies \dfrac{dy}{dx} = \dfrac{ - \sin(x) - 1}{ \{ 1 + \sin(x) \}^{2}}$

$\\\bf \implies \dfrac{dy}{dx} = \dfrac{- \cancel{\{1 + \sin(x) \}}}{ \{ 1 + \sin(x) \}^{ \cancel2}}$

$\\\bf \implies \large{ \boxed{ \bf \dfrac{dy}{dx} = \dfrac{-1}{1 + \sin(x)}}}$

Answer 2

$\rm{f'(x) = - \frac{1}{1 + sin(x)} }$

Explanation :-

Applying quotient rule,

$\rm \: \red {f(x) = \frac{g(x)}{h(x)} }$

$\implies \rm \: \large {f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{h {}^{2}x } ----- (1)}$

In this case we have,

• g(x) = cos(x)
• h(x) = 1+sin(x)
• g'(x) = -sin(x)
• h'(x) = cos(x)

Put all values in eqn. (1) ,

$\implies \rm \: f'(x) = \frac{ - sin \: x(1 + sin \: x) - cos \: x \times cos \: x}{(1 + sin \: x) {}^{2} }$

$\implies \rm \: f'(x) = - \frac{ sin \: x + sin {}^{2} \: x + cos {}^{2} \: x }{(1 + sin \: x) {}^{2} }$

$\implies \rm \: f'(x) = - \frac{sin (x) + 1 }{(1 + sin \: x) {}^{2} }$

$\implies \rm \: f'(x) = - \frac{ \cancel{sin \: x + 1}}{( 1 + sin \: x) {}^{ \cancel{2}} }$

$\implies \rm \: { \boxed{ \rm{ \blue{f'(x) = - \frac{1}{1 + sin \: x} }}}}$

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Formula used :-

• $\rm \: sin {}^{2}x + cos {}^{2} x = 1$