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.
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)`
`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)`
check Approved by eNotes Editorial