Question

find the derivative of the following functions with respect to x 1) f(x) =x/1+x^2. 2) x.sinx​

Answer 1

Answer:

Derivative refers to the value or function obtained after differentiating the given function.

$1. \dfrac{d}{dx} [ \: \dfrac{x}{1+x^2} \: ] = ?\\\\\\\text{ This is of the form:}\\\\\\\implies \dfrac{d}{dx} [ \:\dfrac{u}{v}\: ] \implies \dfrac{(v'u - u'v)}{v^2}$

According to the given question,

• u = x
• v = ( 1 + x² )

Calculating the u' and v' we get:

→ v' = ( 0 + 2x ) = 2x

→ u' = 1

Substituting the values we get:

$\implies \dfrac{ 2x ( x ) - 1 ( 1 + x^2) }{ ( 1 + x^2)^2}\\\\\\\implies \dfrac{ 2x^2 - 1 - x^2 } { 1 + x^4 + 2x^2}\\\\\\\implies \boxed{ \bf{ f'(x) = \dfrac{ x^2 - 1 }{x^4 + 2x^2 + 1}}}$

$2. \: \dfrac{d}{dx} [ \: x.sin\:x \: ]\\\\\\\text{This is of the form:}\\\\\\\implies \dfrac{d}{dx} [uv] = u'v + v'u$

According to the question,

• u = x
• v = sin x

The value of v' and u' are:

• u' = 1
• v' = cos x

Substituting the values we get:

$\implies (cos\:x)(x) + (1)(sin\:x)\\\\\\\implies \boxed{ \bf{ f'(x) = sin\:x + x.cos\:x}}$

These are the required answers.