# How do you differentiate (x^2)arcsin(x^2)?

Apr 3, 2018

$2 x \arcsin \left({x}^{2}\right) + \frac{2 {x}^{3}}{\sqrt{1 - {x}^{4}}}$

#### Explanation:

so you have $\dot{\left(u v\right)} = \dot{u} v + \dot{v} u$

and the derivative of $\arcsin \left(x\right) = \frac{1}{\sqrt{1 - {x}^{2}}}$

so if you take $u = {x}^{2}$ and $v = \arcsin \left({x}^{2}\right)$
$\implies \dot{u} = 2 x \mathmr{and} \dot{v} = \frac{2 x}{\sqrt{1 - {x}^{4}}}$

if you apply the values of $u , v , \dot{u} , \dot{v}$ to $\dot{\left(u v\right)} = \dot{u} v + \dot{v} u$

you'll have

the derivative of ${x}^{2} \arcsin \left({x}^{2}\right)$ is $2 x \arcsin \left({x}^{2}\right) + \frac{2 {x}^{3}}{\sqrt{1 - {x}^{4}}}$

Apr 3, 2018

$f ' \left(x\right) = 2 x \cdot \arcsin \left({x}^{2}\right) + \frac{2 {x}^{3}}{\sqrt{1 - {x}^{4}}}$

#### Explanation:

We have:

$f \left(x\right) = {x}^{2} \cdot \arcsin \left({x}^{2}\right)$.

We use the product rule:

$\frac{d}{\mathrm{dx}} \left[f \left(x\right) \cdot g \left(x\right)\right] = f ' \left(x\right) \cdot g \left(x\right) + f \left(x\right) \cdot g ' \left(x\right)$

$\implies f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left[{x}^{2}\right] \cdot \arcsin \left({x}^{2}\right) + {x}^{2} \cdot \frac{d}{\mathrm{dx}} \left[\arcsin \left({x}^{2}\right)\right]$

$\frac{d}{\mathrm{dx}} \left[f \left(g \left(x\right)\right)\right] = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$.

$\frac{d}{\mathrm{dx}} \left[{x}^{n}\right] = n {x}^{n - 1}$ if $n$ is a constant.

$\frac{d}{\mathrm{dx}} \left[\arcsin \left(x\right)\right] = \frac{1}{\sqrt{1 - {x}^{2}}}$

$\implies f ' \left(x\right) = 2 x \cdot \arcsin \left({x}^{2}\right) + {x}^{2} \cdot \frac{1}{\sqrt{1 - {\left({x}^{2}\right)}^{2}}} \cdot \frac{d}{\mathrm{dx}} \left[{x}^{2}\right]$

$\implies f ' \left(x\right) = 2 x \cdot \arcsin \left({x}^{2}\right) + {x}^{2} \cdot \frac{1}{\sqrt{1 - {x}^{4}}} \cdot 2 x$

$\implies f ' \left(x\right) = 2 x \cdot \arcsin \left({x}^{2}\right) + \frac{{x}^{2} \cdot 2 x}{\sqrt{1 - {x}^{4}}}$

$\implies f ' \left(x\right) = 2 x \cdot \arcsin \left({x}^{2}\right) + \frac{2 {x}^{3}}{\sqrt{1 - {x}^{4}}}$