Question

Find the differentiation of y=cos^-1(ax)?

Answer 1

`y=cos^-1(ax)=>(dy)/(dx)=-1/sqrt(1-(ax)^2)d/(dx)(ax)`
`(dy)/(dx)=-1/sqrt(1-a^2x^2)xxa=(-a)/sqrt(1-a^2x^2)`
Note: `color(red)(d/(dX)(cos^-1X)=-1/sqrt(1-X^2)`

Explanation:

Here,

`y=cos^-1(ax)`

Let, `u=ax=>(du)/(dx)=a`

`So, y=cos^-1u=>(dy)/(du)=-1/sqrt(1-u^2)`

`"Using "color(blue)"Chain Rule"`,

`color(blue)((dy)/(dx)=(dy)/(du)*(du)/(dx)`

`:.(dy)/(dx)=-1/sqrt(1-u^2)xxa=(-a)/sqrt(1-u^2),where, u=ax`

`=>(dy)/(dx)=(-a)/sqrt(1-(ax)^2)`

`=>(dy)/(dx)=(-a)/sqrt(1-a^2x^2)`