# Find the derivative using the derivative rules ?

Dec 8, 2017

color(purple)(10xsin^-1(4x) + (20x^2)/sqrt(1-16x^2)

#### Explanation:

There's 3 rules you need to use here:

When taking derivatives, you always want to work outside-in. The outermost rule here is the product rule , so you'd use that.

Video, in case you need it:

The general rule is:

$\frac{d}{\mathrm{dx}} \left[\textcolor{red}{f \left(x\right)} \cdot \textcolor{b l u e}{g \left(x\right)}\right] = \textcolor{red}{\frac{d}{\mathrm{dx}} \left[f \left(x\right)\right]} \cdot \textcolor{b l u e}{g \left(x\right)} + \textcolor{b l u e}{\frac{d}{\mathrm{dx}} \left[g \left(x\right)\right]} \cdot \textcolor{red}{f \left(x\right)}$

So:
$\frac{d}{\mathrm{dx}} \left(5 {x}^{2} {\sin}^{-} 1 \left(4 x\right)\right) = \textcolor{red}{\frac{d}{\mathrm{dx}} \left(5 {x}^{2}\right)} \cdot \textcolor{b l u e}{{\sin}^{-} 1 \left(4 x\right)} + \textcolor{b l u e}{\frac{d}{\mathrm{dx}} {\sin}^{-} 1 \left(4 x\right)} \cdot \textcolor{red}{5 {x}^{2}}$

That gives you two derivatives you need to evaluate.

To evaluate the first of these two derivatives, you'll need to use the power rule.

Video, in case you need it:

The general rule is:

$\frac{d}{\mathrm{dx}} \textcolor{g r e e n}{{x}^{a}} = \textcolor{g r e e n}{a {x}^{a - 1}}$

So, what you'd have is:

d/dxcolor(green)(5x^2) = color(green)(2(5x^(2-1)) = color(green)(10x)

To evaluate the second of these two derivatives, you'll need to employ the chain rule.

Video, in case you need it:

The general rule is:

$\frac{d}{\mathrm{dx}} \textcolor{red}{f \left(\textcolor{b l u e}{g \left(x\right)}\right)} = \textcolor{red}{f ' \left(g \left(x\right)\right)} \cdot \textcolor{b l u e}{g ' \left(x\right)}$

So:

$\frac{d}{\mathrm{dx}} \textcolor{red}{{\sin}^{-} 1} \left(\textcolor{b l u e}{4 x}\right) = \textcolor{red}{\frac{d}{\mathrm{dx}} \left({\sin}^{-} 1 \left(4 x\right)\right)} \cdot \textcolor{b l u e}{\frac{d}{\mathrm{dx}} \left(4 x\right)}$

color(red)(1/sqrt[1-(4x)^2] * color(blue)(4)

= color(orange)(4/sqrt(1-16x^2)

Now, we just put it all together:

d/dx(5x^2sin^-1(4x)) = color(green)(10x)*color(blue)(sin^-1(4x)) + color(blue) color(orange)(4/sqrt(1-16x^2)*color(red)(5x^2)

Simplify, and it all boils down to:

 = color(purple)(10xsin^-1(4x) + (20x^2)/sqrt(1-16x^2)

Hope that helps :)