Question

if y=cos(1/x²)then dy/dx is​

Answer 1

Given :

y = cos(1/x²)

• Apply chain rule here

$\sf \dfrac{dy}{dx} = \dfrac{dcos( \dfrac{1}{x^{2} })}{d(\dfrac{1}{x^{2}})} \: . \: \dfrac{d( \dfrac{1}{x^{2}})}{dx}$

• Differentiate with help of quotient rule

$\sf \dfrac{dy}{dx} = - sin \dfrac{1}{x^{2}} \: . \: \left[\dfrac{x^{2} . \dfrac{d1}{dx }{ - 1. \dfrac{dx^{2}}{dx}}}{(x^{2})^{2} } \right]$

• As we know that differentiation some basic functions
• cos x = (-sin x)
• constant term = 0
• x^n = na^n-1

$\sf \dfrac{dy}{dx} = - sin \dfrac{1}{x^{2}} \: . \: \left[\dfrac{x^{2} .0 - 2x}{(x^{2})^{2} } \right]$

• After applying simplify it

$\sf \dfrac{dy}{dx} = - sin \dfrac{1}{x^{2}} \: . \: \left[\dfrac{ - 2x}{x^{4}}\right]$

$\sf \dfrac{dy}{dx} = sin \dfrac{ 2x}{x^{6}}$

• Cancel numerator of x with denominator of x⁶.

$\sf \dfrac{dy}{dx} = sin \dfrac{ 2}{x^{5}}$