Question

Find the derivative of $f\left(x\right)=x^2$.

Answer 1

Given : $f\left(x\right)=x^2$
We know that $f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{(x+h{)}^2-x^2}{h}$
$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{x^2+h^2+2xh-x^2}{h}$
$\Rightarrow f^{\prime \prime }\left(x\right)=\underset{h\to 0}{lim}\frac{h^2+2xh-x^2+x^2}{h}$
$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{h(h+2x)+0}{h}$
$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{h(h+2x)}{h}$
$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}(h+2x)$
$\Rightarrow f^{\prime }\left(x\right)=0+2x$
$\Rightarrow f^{\prime }\left(x\right)=2x$
Hence, $f^{\prime }\left(x\right)=2x$