Finding derivative of a function

[GiTS]

Homework Statement

Use the definition of the derivative lim_h->0 f(x+h)-f(x)
_______________________________________h
to show that the derivative of f(x)=x³ is f`(x)=3x²

limits and derivatives are very confusing to me. I got lost after the first two steps and I'm not sure if they are correct. I think the next step is to get rid of h, but I don't know how to go about it. Any help or suggestions would be much appreciated. Thanks!
p.s. the "_____" don't mean anything, I just couldn't figure out how to do a space.

Homework Equations

The Attempt at a Solution

(x+h)³-x³
______h
= (x+h)(x+h)(x+h)-x³
______h

 

[sadean1s]

Take the exponent and multiply it by the coefficient in this case 1. You get 3x. Then subtract 1 from the exponent, 3, and you get 2. Giving you 3x²

3(x)³
3x^3-1
=3x²

 

[cristo]

The Attempt at a Solution

(x+h)³-x³
______h
= (x+h)(x+h)(x+h)-x³
______h

Why not try expanding out the numerator?

Take the exponent and multiply it by the coefficient in this case 1. You get 3x. Then subtract 1 from the exponent, 3, and you get 2. Giving you 3x²

3(x)³
3x^3-1
=3x²

As the OP states, he is required to calculate this using the definition on the derivative. Also, please note in future that we do not give out full solutions in the homework forums, only tutorial advice.

 

[GiTS]

I can expand the first two parts to get (x²+2xh+h²)(x+h)
I'm not really sure what how to multiply x and 2xh or x and h². I don't think you can, can you?
Youd have to write it as x³+(x)(2xh)+(h)(2xh)+(x)(h²)+(h)(x²)+h³-x³/h
correct?

then take out the x³ and -x³

somehow I have to get rid of those xh's and h's and x².

 

[Mark44]

I can expand the first two parts to get (x²+2xh+h²)(x+h)
I'm not really sure what how to multiply x and 2xh or x and h². I don't think you can, can you?
Youd have to write it as x³+(x)(2xh)+(h)(2xh)+(x)(h²)+(h)(x²)+h³-x³/h
correct?

then take out the x³ and -x³

somehow I have to get rid of those xh's and h's and x².

I can multiply them, even if you can't. To multiply (x²+2xh+h²) by (x + h), multiply all the terms in the first expression by x, and then multiply the terms in the first expression by h.

For example, one of the terms in the first expression is 2xh. When you multiply this by x, you get 2x²h, and when you multiply the same term by h, you get 2xh². You'll end up with six terms, and sum of the terms are like expressions (they differ only in their coefficients). Group together the like terms and add their coefficients. You should end up with four terms.

Next, subtract the x³. All the terms that are left will have at least one factor of h, so you can factor it out and cancel with the h in the denominator.

At that point you're ready to take the limit as h --> 0, and you should end up with the derivative you're looking for.
Mark