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Math Help - Find f'(x) using the definition of derivative

  1. #1
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    Find f'(x) using the definition of derivative

    Hey guys/gals, I have a midterm coming up Wends for my Calc class and we were given a practice test and there is a problem on here that I am struggling coming up with the correct answer. Thanks in advance you guys have been a huge help this semester. Below is the problem.

    Use teh definition f'(x) = lim as h--->0 f(x+h)-f(x)/h f(x)= 3x^4 - 9x^3 + 4

    so far I have gotten this far haha. j/k I have done this so far.

    lim h__>0 [3(x+h)^4 - 9(x+h)^3 +4] - (3x^4 - 9x^3 + 4] / h

    The 4's cancel out but then I need to break down the x+h's and that is where I am losing it. Can someone show me the correct way. I would love to just use the power rule but my professor said we need to work them out. I know this is a long problem but I need to see the steps so I can compare with mine to find my mistake. Thanks again.
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  2. #2
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    Re: Find f'(x) using the definition of derivative

    Quote Originally Posted by psilver1 View Post
    Hey guys/gals, I have a midterm coming up Wends for my Calc class and we were given a practice test and there is a problem on here that I am struggling coming up with the correct answer. Thanks in advance you guys have been a huge help this semester. Below is the problem.

    Use teh definition f'(x) = lim as h--->0 f(x+h)-f(x)/h f(x)= 3x^4 - 9x^3 + 4

    so far I have gotten this far haha. j/k I have done this so far.

    lim h__>0 [3(x+h)^4 - 9(x+h)^3 +4] - (3x^4 - 9x^3 + 4] / h

    The 4's cancel out but then I need to break down the x+h's and that is where I am losing it. Can someone show me the correct way. I would love to just use the power rule but my professor said we need to work them out. I know this is a long problem but I need to see the steps so I can compare with mine to find my mistake. Thanks again.
    For a problem like this the binomial theorem will be very useful

    (a+b)^n=\sum_{i=0}^{n}\binom{n}{i}a^{n-i}b^{i}

    We have

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

    f'(x)=\lim_{h \to 0}\frac{[3(x+h)^4-9(x+h)^3+4]-[3x^4-9x^3+4]}{h}

    Now this is where the binomial theorem is useful

    (x+h)^4=x^4+4x^3h+6x^2h^2+4xh^3+h^4

    (x+h)^3=x^3+3x^2h+3xh^2+h^3

    Now just plug and chug.
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  3. #3
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    Re: Find f'(x) using the definition of derivative

    I see how you did the binomial theorem but could you give me the definition so that I may learn it? or better yet an explanation of how you come up with those terms
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