# ^(a ) find f^-1(x)

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### 4 Answers

You need to find the inverse of the function `f(x) = e^(sin^(-1) x)` using logarithmation, such that:

`y = e^(sin^(-1) x) => ln y = ln e^(sin^(-1) x)`

Uisng logarithmic identity ` ln a^b = b*ln a` yields:

`ln y = (sin^(-1) x)*ln e`

Since` ln e = 1` yields:

`ln y = (sin^(-1) x) `

Taking sine function both sides, yields:

`sin ln y = sin((sin^(-1) x) ) => x = sin ln y`

**Hence, evaluating the inverse function, yields **`f^(-1)(x) = sin ln x.`

its a composite function:

`f(z)= e^(az)` `z=g(y)=1/y` `y=h(x)=senx`

so:

`f'(x)= f'(z)g'(y)h'(x)` `=ae^(az)(-1/y^2)cosx`

substiuing y in x function value:

`f'(x)=ae^(az)(-cosx/(sen^2x))`

substituing y value of function x in value to z:

`f'(x)=-ae^(a/(senx))(cosx)/(sen^2x)`

You need to find the inverse of the function using logarithmation, such that:

Uisng logarithmic identity yields:

Since yields:

Taking sine function both sides, yields:

Hence, evaluating the inverse function, yields

i am seaching for differentation of f(x) not inverse'

f(x)=e^(a`sin^-1` (x))

find `f^l(x)`