Find the derivative of f(x)=x^(8sin5x)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use the properties of logarithms to find the derivative of the function such that:

`ln f(x) = ln x^(8sin 5x)`

You should use the notation `y = f(x)`  and the power property of logarithms such that:

`ln y = (8sin 5x) * ln x`

You need to differentiate both sides with respect to x such that:

`y'*(1/y) = (8sin 5x)' * ln x + (8sin 5x) * (ln x)' ` (expand the right side using the product rule)

`y'*(1/y) =40 cos 5x* ln x + (8sin 5x)/x`

Multiplying by y both sides

`y' = y*(40 cos 5x* ln x + (8sin 5x)/x)`

`` You need to substitute y by its equation `y = x^(8sin 5x)`

`y' = x^(8sin 5x)*(40 cos 5x* ln x + (8sin 5x)/x)`

Hence, evaluating the derivative of the function f(x) yields `f'(x) = x^(8sin 5x)*(40 cos 5x* ln x + (8sin 5x)/x).`

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