### Home > APCALC > Chapter 3 > Lesson 3.2.3 > Problem3-76

3-76.

Compare three different methods to find a derivative of $f(x) = 2x^3 − x$. Homework Help ✎

1. Use the definition of a derivative.

$f'(x)=\lim\limits_{h\rightarrow 0}\frac{(2(x+h)^{3}-(x+h))-(2x^{3}-x)}{h}$

Expand the numerator.

Combine like terms to simplify the numerator.

Factor out an $h$.

Simplify the fraction by 'cancelling out' the $h$.

Evaluate the limit as $h \rightarrow 0$.

3. Use your graphing calculator to graph $f^\prime (x)= \frac{f(x+h)-f(x)}{h}$ for $h = 0.01$. Does the graph match that of your answer from part (a)?
Sketch $y=\frac{(2(x+0.01)^{3}-(x+0.01))-(2x^{3}-x)}{0.01}$
also sketch the $f^\prime (x)$ you found in part (a) and part (b).
The two graphs should be close, but not identical since part (c) uses a finite value for $h$.