Question

solve x^3 - 3x + 5 using the definition of the derivative

Answer 1

Differentiate or solving this equation?

Answer 2

differentiate

Answer 3

Definition of derivatives:
\[
f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}
\]

Answer 4

yes but when i plug it in then the answer becomes complicated i get to

Answer 5

\[[(x+h)^3 - 3(x+h) + 5 * x^3 -3x +5]/h \]

Answer 6

\[
\lim_{h\to 0}\frac{f(x+h)-f(x)}{h},\text{ not}\lim_{h\to 0}\frac{f(x+h)\times f(x)}{h}
\]

Answer 7

oh! let me retry it!

Answer 8

i get to the point where i do (x+h)^3 and i dont really know what it comes out to i end up with x^3 + 2x^2h + h^2x + hx^2 + 2xh^2 + h^3 and i dont know how to simplify this

Answer 9

You can directly use binomial theorem to expand \((x+h)^3\).
Binomial theorem:
\[
(x+y)^n=\sum_{k=0}^n\binom{n}{k}x^ky^{n-k}
\]
I don't know how to explain the theorem though.

Answer 10

i dont understand that theorem either

Answer 11

There is another way, since it's a small power you could use pascal's triangle.

Answer 12

Just to get the constants that is.

Answer 13

With lower powers, multiplying longhand is often the fastest, simplest way.

Answer 14

Try Khan Academy?
https://www.khanacademy.org/math/algebra2/polynomial_and_rational/binomial_theorem/v/binomial-theorem

Answer 15

This what longhand multiplication looks like

\(\large \displaystyle \begin{align} x &+ h \cr x &+ h \cr \hline hx &+ h^2 \cr x^2 + hx& \cr\hline x^2 + 2h&x + h^2 \cr &x + h \cr \hline hx^2 +2h^2&x + h^3 \cr x^3 +2hx^2 +h^2x& \cr \hline x^3 + 3hx^2 +3h^2&x + h^3 \end{align}\)

Answer 16

thank you for explaining it clearly. i understand how to do it now thanks everyone