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