Differentiate the function w(x)=sin^-1 (x/4)

I know that the derivate of sin^-1 is 1/sqrt(1-x^2) but I don't know where to go from there.

Expert Answers

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Consider the following situation.

y = f(x)

P = f(y)

y is a function of x and P is a function of y.

Directly we cannot find any combination with P and x.

If we want (dP)/dx then;

`(dP)/dx = (dP)/dy*(dy)/dx`

Since P is a function of y and y is a function of x we can evaluate (dP)/dx.

This is known as derivative of function of function.

`w(x) = sin^(-1)(x/4)`

let;

`t= x/4`

`w(x) = sin^(-1)t`

`(dw(x))/dx = (dw(x))/dt*(dt)/dx`

`(dw(x))/dt = 1/sqrt(1-t^2) = 1/sqrt(1-(x/4)^2) = 4/sqrt(16-x^2)`

`(dt)/dx = 1/4`

`(dw(x))/dx = 4/sqrt(16-x^2)*(1/4) = 1/sqrt(16-x^2)`

`(dw(x))/dx = 1/sqrt(16-x^2)`

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