Question

Basic definition-of-derivative calculations: Find the derivatives of the functions that follow, using (a) the $h\to 0$definition of the derivative and (b) the $z\to x$definition of the derivative.

$f\left(x\right)=x^3$

Short answer

$f\text{'}\left(x\right)=3x^2$

Step-by-step solution

Given Information

The given expression is$f\left(x\right)=x^3$

Simplification

Using (a)

$f^{\text{'}}\left(x\right)=\underset{h\to 0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}$

$f^{\text{'}}\left(x\right)=\underset{h\to 0}{lim}\frac{{\left(x+h\right)}^3-x^3}{h}$

role="math" localid="1650970063541" $f^{\text{'}}\left(x\right)=\underset{h\to 0}{lim}\frac{x^3+h^3+3x^{2}h+3x{h}^2-x^3}{h}$

role="math" localid="1650970073601" $f^{\text{'}}\left(x\right)=\underset{h\to 0}{lim}\frac{h\left(h^2+3x^2+3xh\right)}{h}$

role="math" localid="1650970082470" $f^{\text{'}}\left(x\right)=\underset{h\to 0}{lim}{h}^2+3x^2+2xh$

Applying limit

$f\left(x\right)=2x^2$

Using (b)

role="math" localid="1650970131635" $f^{\text{'}}\left(x\right)=\underset{z\to x}{lim}\frac{f\left(z\right)-f\left(x\right)}{z-x}$

role="math" localid="1650969972120" $f^{\text{'}}\left(x\right)=\underset{z\to x}{lim}\frac{z^3-x^3}{z-x}$

role="math" localid="1650969988908" $f^{\text{'}}\left(x\right)=\underset{z\to x}{lim}\frac{\left(z-x\right)\left(z^2+x^2+2zx\right)}{z-x}$

$f^{\text{'}}\left(x\right)=\underset{z\to x}{lim}\left(z^2+x^2+2zx\right)$

Applying limit

$f^{\text{'}}\left(x\right)=3x^2$