Find f ‘(−3), if f(x) = (2x^2 − 7x)(−x^2 − 7). Round your answer to the nearest integer. Use the hyphen symbol, -, for negative values.

Question

Find f ‘(−3), if f(x) = (2x^2 − 7x)(−x^2 − 7). Round your answer to the nearest integer. Use the hyphen symbol, -, for negative values.

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    0
    2021-09-09T16:41:28+00:00

    Answer: f'(-3)=538

    Step by step:

    To get the derivative you can either multiply out the product (which I did below), or apply the formula for the derivative of a product of two functions. Either way you will obtain the same result, of course.

    f(x) = (2x^2-7x)(-x^2-7)=-2 x^4 + 7 x^3 - 14 x^2 + 49 x\\f'(x) = -8x^3 +21x^2 - 28x +49\\f'(-3)=538

    0
    2021-09-09T16:41:30+00:00

    use product rule

    \frac{d}{dx} g(x)h(x)=g'(x)h(x)+g(x)h'(x) (where the ‘ symbol is derivitive with respect to x, (just using Leibniz notation)

    also remember the power rule: \frac{d}{dx} x^n=nx^{n-1}

    and sum rule, \frac{d}{dx} (g(x)+h(x))=\frac{d}{dx}g(x)+\frac{d}{dx}h(x)

    so first find the derivitive then evaluate it

    if we say that 2x^2-7x=g(x) and -x^2-7=h(x)

    setup:

    find g'(x) and h'(x)

     g'(x)=2*2x^1-7*1x^0=4x-7*1=4x-7

    h'(x)=2*(-x^1)-0=-2x

    so f'(x)=g'(x)h(x)+g(x)h'(x)=(4x-7)(-x^2-7)+(2x^2-7x)(-2x)

    evaluate f'(-3)

    f'(-3)=(4(-3)-7)(-(-3)^2-7)+(2(-3)^2-7(-3))(-2(-3))

    f'(-3)=(-19)(-16)+(39)(6)

    f'(-3)=538

    answer: f'(-3)=538

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