# Find the derivative of f(x)= e^(cosx)/x

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should notice that more than two functions form f(x) and f(x) is a rational function, hence you need to differentiate with respect to x, using two rules of derivatives, thus you need to use quotient rule since the function is rational and chain rule since the function `e^(cos x)`  is composed.

`f'(x) = ((e^(cosx))'*x - e^(cosx)*(x)')/x^2`

`f'(x) = ((e^(cosx))*(cos x)'*x - e^(cosx)*1)/x^2`

`f'(x) = ((e^(cosx))*(-sin x)*x - e^(cosx))/x^2`

You need to factor out `- e^(cosx)`  such that:

`f'(x) = - e^(cosx)*(x*sin x + 1)/x^2`

Hence, evaluating the derivative of function yields f'`(x) = - e^(cosx)*(x*sin x + 1)/x^2` .

justaguide | College Teacher | (Level 2) Distinguished Educator

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The function f(x) = `e^(cosx)/x`

f'(x) = `((e^(cos x))'*x - e^(cos x)*x')/x^2`

=> `(-e^cos x*sin x*x - e^cos x)/x^2`

=> `((-x*sin x - 1)*e^cos x)/x^2`

The derivative of `f(x) = e^(cosx)/x` is `f'(x) = ((-x*sin x - 1)*e^cos x)/x^2`