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Math Help - Differentiate the following function

  1. #1
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    Differentiate the following function

    f(x) = 5x^3 - 6x + 8

    f ' (x) =
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  2. #2
    Senior Member topher0805's Avatar
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    f'(x) = limit as h approaches 0 of \frac {f(x+h) - f(x)}{h}

    Can you solve it now?
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  3. #3
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    Is this right?

    5(x+h)^3-6(x+h)+8-(5x^3-6x+8)/h = 5x^3+5h^3-6x+6h+8-5x^3+6x-8/h =

    5h^3+6h/h = h(5^3+6)/h = 131
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  4. #4
    Senior Member topher0805's Avatar
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    Looks good to me.
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  5. #5
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    I put this in and its coming up wrong, can u help me find where i went wrong
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  6. #6
    Senior Member topher0805's Avatar
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    Ok let me walk you through it.

    f'(x) = \lim_{h \to 0} \frac {f(x + h) - f(x)}{h}

    So plug in 5x^3 - 6x + 8 to get:

    \lim_{h \to 0} \frac {5(x + h)^3 - 6(x + h) + 8 - (5x^3 - 6x + 8)}{h}

    Now use your factoring skills:

    \lim_{h \to 0} \frac {5(x+h)(x^2 + 2xh + h^2) - 6x - 6h + 8 - 5x^3 + 6x - 8}{h}

    Becomes:

    \lim_{h \to 0} \frac {5x^3 + 10x^2h + 5xh^2 + 5x^2h + 10xh^2 + 5h^3 - 6x - 6h + 8 - 5x^3 + 6x - 8}{h}

    Combine like terms:

    \lim_{h \to 0} \frac {15x^2h + 15xh^2 + 5h^3 - 6h}{h}

    Factor out an h:

    \lim_{h \to 0}15x^2 + 15xh + 5h^2 - 6

    As h approaches 0, so do 15xh and 5h^2, so we have that:

    f'(x) = \lim_{h \to 0} \frac {f(x + h) - f(x)}{h} = 15x^2 - 6
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