Question

differentiate the following with respect to x. COSCOS X/
x+sinsin x​

Answer 1

AnswEr :

Given Expression,

$\sf \: f(x) = \dfrac{cos(cos \: x)}{x + sin(sin \: x)}$

The above expression is of the form u/v

Derivative of u/v will be of the form :

$\sf \: f'(x) = \dfrac{u'v - v' u}{v {}^{2} }$

Now,

$\longrightarrow \: \sf \: f'(x) = \dfrac{d \bigg(cos(cos \: x) \bigg) \big(x + sin(sin \: x) \big) - d \bigg(x + sin(sin \: x) \bigg) \big(cos(cos \: x) \big)}{ \big(x + sin(sin \: x) \big) {}^{2} } \\ \\ \longrightarrow \: \sf \: f'(x) = \dfrac{ - sin(cos \: x) \big(x + sin(sin \: x) \big) \dfrac{d(cos \: x)}{dx} - \bigg(1 + cos(sin \: x) \times \dfrac{d(sin \: x)}{dx} \bigg) \big(cos(cos \: x) \big)}{ \big(x + sin(sin \: x) \big) {}^{2} } \\ \\ \longrightarrow \boxed{ \boxed{ \sf \: f'(x) = \dfrac{sin(cos \: x).sin \: x \big(x + sin(sin \: x) \big) - \big(1 + cos \: x.cos(sin \: x) \big) \big(cos(cos \: x) \big)}{ \big(x + sin(sin \: x) \big) {}^{2} } }}$

• Derivative of cos x is - sin x

• Derivative of sin x is cos x