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Let f(x)=(-9x^2-4)^6(7x^2-7)^14 what is f'(x)?

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adb2188 | (Level 3) eNoter

Posted February 23, 2013 at 9:43 PM via web

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Let f(x)=(-9x^2-4)^6(7x^2-7)^14

what is f'(x)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted February 24, 2013 at 12:29 PM (Answer #1)

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You need to evaluate the derivative of the function, hence, you need to use the product rule and the chain rule, such that:

`f'(x) = ((-9x^2-4)^6)'(7x^2-7)^14 + (-9x^2-4)^6((7x^2-7)^14)'`

`f'(x) = 6(-9x^2-4)^5(-9x^2-4)'(7x^2-7)^14 + 14(-9x^2-4)^6*(7x^2-7)^13*(7x^2-7)'`

`f'(x) = 6(-9x^2-4)^5(-18x)(7x^2-7)^14 + 14(-9x^2-4)^6*(7x^2-7)^13*(14x)`

`f'(x) = -108x*(-9x^2-4)^5*(7x^2-7)^14 + 196x*(-9x^2-4)^6*(7x^2-7)^13`

You may factor out `x*(-9x^2-4)^5*(7x^2-7)^13` such that:

`f'(x) = x*(-9x^2-4)^5*(7x^2-7)^13*(-108(7x^2-7) + 196(-9x^2-4))`

`f'(x) = x*(-9x^2-4)^5*(7x^2-7)^13*(-756x^2 + 756 - 1764x^2 - 784)`

`f'(x) = x*(-9x^2-4)^5*(7x^2-7)^13*(-2520x^2 - 28)`

`f'(x) = -28x*(-9x^2-4)^5*(7x^2-7)^13*(90x^2 + 1)`

Hence, evaluating the derivative of the given function yields `f'(x) = -28x*(-9x^2-4)^5*(7x^2-7)^13*(90x^2 + 1).`

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