Question

Differentiate from first principle:

$\left(xi\right)(x+2{)}^3$

Answer 1

Given:

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

The derivative of a function $f\left(x\right)$ is defined as:

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

Putting $f\left(x\right)$ in the above expression, we get:

$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{(x+h+2{)}^3-(x+2{)}^3}{h}$

$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{\left[(x+2{)}^3+h^3+3h(x+2)\{(x+2)+h\}\right]-(x+2{)}^3}{h}$

$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{h^3+3h(x+2)\left\{\right(x+2)+h\}}{h}$

$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}[h^2+3(x+2\left)\right\{(x+2)+h\left\}\right]$

$\Rightarrow f^{\prime }\left(x\right)=3(x+2)(x+2+0)$

$\Rightarrow f^{\prime }\left(x\right)=3(x+2{)}^2$

Therefore, the derivative of $(x+2{)}^3\text{is}3(x+2{)}^2.$