Question

Find the derivative of cos⁻¹x w.r. to `sqrt(1 - x^2)`

Answer 1

Let u = cos⁻¹x 

Differentiating w. r. t. x, we get

`("d"u)/("d"x) = "d"/("d"x)(cos^-1 x)`

= `(-1)/sqrt(1 - x^2)`

Let v = `sqrt(1 - x^2)`

Differentiating w. r. t. x, we get

`("dv")/("d"x) = "d"/("d"x)(sqrt(1 - x^2))`

= `1/(2sqrt(1 - x^2))*"d"/("d"x)(1 - x^2)`

= `1/(2sqrt(1 - x^2))*(-2x)`

= `(-x)/sqrt(1 - x^2)`

∴ `("d"u)/("dv") = (("d"u)/("d"x))/(("dv")/("d"x))`

= `(-1)/((sqrt(1 - x^2))/((-x)/(sqrt(1 - x^2))`

= `1/x`