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.

Re: Find f'(x) using the definition of derivative

Quote:

Originally Posted by

**psilver1** 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

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

We have

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

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

Now this is where the binomial theorem is useful

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

$\displaystyle (x+h)^3=x^3+3x^2h+3xh^2+h^3$

Now just plug and chug.

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