`y = sin(x)/(x^4)` Use the Product Rule or the Quotient Rule to find the derivative of the function.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the derivative of the given function and since the function is a quotient of two functions, then you must use the quotient rule, such that:

`f'(x) = ((x^4)(sin x)' - (x^4)'(sin x))/(x^8)`

`f'(x) = (x^4*cos x - 4x^3*sin x)/(x^8)`

Factoring out `x^3 ` yields:

`f'(x) = x^3(x*cos x - 4*sin x)/(x^8)`

Reducing like terms yields:

`f'(x) = (x*cos x - 4*sin x)/(x^5)`

Hence, evaluating the derivative of the function, using the product rule, yields `f'(x) = (x*cos x - 4*sin x)/(x^5).`