Question

Find the derivative at $x=2$ of the function $f\left(x\right)=3x$

Answer 1

Given : $f\left(x\right)=3x$
We know that,
$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{3(x+h)-3\left(x\right)}{h}$
Putting $x=2$
$\Rightarrow f^{\prime }\left(2\right)=\underset{h\to 0}{lim}\frac{3(2+h)-3\left(2\right)}{h}$
$\Rightarrow f^{\prime }\left(2\right)=\underset{h\to 0}{lim}\frac{6+3h-6}{h}$
$\Rightarrow f^{\prime }\left(2\right)=\underset{h\to 0}{lim}\frac{3h+0}{h}$
$\Rightarrow f^{\prime }\left(2\right)=\underset{h\to 0}{lim}\frac{3h}{h}$
$\Rightarrow f^{\prime }\left(2\right)=\underset{h\to 0}{lim}3$
$\Rightarrow f^{\prime }\left(2\right)=3$
Hence the derivative of the function $f\left(x\right)$ at $x=2$ is $3$.