Question

The derivative of a function fat a number ais

$f\text{'}\left(a\right)=\underset{h\to 0}{\lim }\frac{\begin{array}{c}\_\_\_\end{array}-\_\_\_}{\_\_\_}$

if the limit exists. The derivative$f\text{'}\left(a\right)$is theof the tangent line to the curve$y=f\left(x\right)$at the point$(\_\_,\_\_)$.

Short answer

The derivative of a function is$f\text{'}\left(a\right)=\frac{f(a+h)-f\left(a\right)}{h}$ .

The derivative$f\text{'}\left(a\right)$ is the slopeof the tangent line to the curve at the point$(a,f(a\left)\right)$ .

Step-by-step solution

Step 1. Given information.

The function here given is,

$y=f\left(x\right)$

Step 2. Formula used.

We have seen that the slope of the tangent line to the curve$y=f\left(x\right)$at the point$(a,f(a\left)\right)$can be written as,

$\underset{h\to 0}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$.

Therefore, the derivative of a function f at a number a is$f\text{'}\left(a\right)=\underset{h\to 0}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$ if limit exists. The derivative$f\text{'}\left(a\right)$ is the slope of the tangent line to the curve$y=f\left(x\right)$ at the point$(a,f(a\left)\right)$