Let f(x)= ((x^6)*((x-8)^2))/(((x^2)+4)^7) Use logarithmic differentiation to determine the derivative. f'(x)=? and f'(8)=?   

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You do not need to use logarithmic differentiation since the variable x is not found to exponent, hence you may use product, quotient, power and chain rules such that:

`f'(x) = (((x^6)*(x-8)^2)'*((x^2)+4)^7) - ((x^6)*(x-8)^2)*((x^2)+4)^7)')/(((x^2)+4)^14)`

Use product rule, power rule and chain rule to differentiate `((x^6)*(x-8)^2)'`  with respect to x such that:

`((x^6)*(x-8)^2)' = 6x^5*(x-8)^2 - 2x^6*(x-8)`

`((x^6)*(x-8)^2)' = 2x^5*(x-8)(3x - 24 - x)`

`((x^6)*(x-8)^2)' = 4x^5*(x - 8)(x -12)`

`((x^6)*(x-8)^2)' = 4x^5*(x^2 - 20x + 96)`

`((x^6)*(x-8)^2)' = 4x^7 - 80x^6 + 384x^5`

`f'(x) = (4x^5*(x - 8)(x - 12) - 14x^7*(x-8)^2*(x^2+7)^6)/(((x^2)+4)^14)`

You may factor out `2x^5*(x - 8) ` such that:

`f'(x) = (2x^5*(x - 8)(2(x - 12) - 7x^2*(x-8)*(x^2+7)^6))/(((x^2)+4)^14)`

You need to substitute 8 for x in f'(x) such that:

`f'(8) = (2*8^5*(8 - 8)(2(8 - 12) - 7x^2*(8-8)*(8^2+7)^6))/(((8^2)+4)^14)`

`f'(8) = 0/(((8^2)+4)^14) = 0`

Hence, evaluating derivative of function yields `f'(x) = (2x^5*(x - 8)(2(x - 12) - 7x^2*(x-8)*(x^2+7)^6))/(((x^2)+4)^14)`  and `f'(8) = 0` , thus, the function has a critical value at `x=8` .

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