Question

question 23
use the limit definition of the derivative to find $f(x)$ when $f(x)=7x^2$.
$\bigcirc$ $f(x)=14x + 7h$
$\bigcirc$ $f(x)=14x$
$\bigcirc$ $f(x)=2x$
$\bigcirc$ $f(x)=0$

Explanation:

Step1: Recall limit definition of derivative

The limit definition of the derivative is:
$$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$

Step2: Substitute $f(x)=7x^2$ into formula

First find $f(x+h)=7(x+h)^2=7(x^2+2xh+h^2)=7x^2+14xh+7h^2$. Then:
$$\frac{f(x+h)-f(x)}{h} = \frac{7x^2+14xh+7h^2 - 7x^2}{h}$$

Step3: Simplify the numerator

Cancel $7x^2$ terms:
$$\frac{14xh+7h^2}{h}$$

Step4: Factor and cancel $h$

Factor $h$ from numerator:

$$\frac{h(14x+7h)}{h} = 14x+7h \quad (h eq 0)$$

Step5: Evaluate the limit as $h\to0$

Substitute $h=0$:
$$\lim_{h \to 0} (14x+7h) = 14x$$

Answer:

$f'(x)=14x$