Differentiate the function s(x)=sec^-1 (x/2) I know that the derivative of sec^-1 is 1/x sqrt (x^2-1) but I don't know what to do from there.

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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.

 

`s(x) = sec^(-1)(x/2)`

 

Let;

`t = x/2`

 

`s(x) = sec^(-1)t`

 

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

 

`(ds(x))/dt = 1/(t*sqrt(^2-1)) = 1/(x/2*sqrt(x^2/4-1)) = 4/(xsqrt(4-x^2))`

 

`dt/dx = 1/2`

 

`(ds(x))/dx = 4/(xsqrt(4-x^2))*1/2 = 2/(xsqrt(4-x^2))`

 

`(ds(x))/dx = 2/(xsqrt(4-x^2))`

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