Thread: Finding a derivative using first principles

I'm working on this question:

if f(x) = x^3, then the derivative is 3x^2

I have to prove this using the definition of the derivative. I got as far as this:



I'm unsure of how to simplify it in order to find the limit as h -> 0.

lim h --> 0 of x^3 = [(x+h)^3 - x^3]/h
= [(x+h)(x+h)(x+h) - x^3]/h
= [x^3 + x^2 h + 2 x^2 h + 2 x h^2 + x h^2 + h^3 - x^3]/h
cancel the x^3 and the h's with the one in the bottom
= x^2 + 2 x^2 + 2 x h + x h + h^2
plug in 0 for h
= x^2 + 2 x^2
combine like terms
= 3 x^2

I hope that helps

Ahh, thanks! The problem was with my simplification of (x+h)^2

I have a similar question:

Find the derivative of using first principles. Once again, I'm having trouble simplifying. :/

I keep getting when it should be

Originally Posted by Glitch

I have a similar question:

Find the derivative of using first principles. Once again, I'm having trouble simplifying. :/

I keep getting when it should be

Please show all your working. (And in future please post a new question in a new thread).

Working:









As h -> 0,



Which is incorrect.

Originally Posted by Glitch

Working:



Mr F says: This is wrong. All that follows from it will therefore be wrong. It should be .





As h -> 0,



Which is incorrect.

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