# How do you differentiate y=arcsin(3x)/x?

Jan 15, 2018

$y ' = \frac{3 x - \arcsin \left(3 x\right) \sqrt{1 - 9 {x}^{2}}}{{x}^{2} \sqrt{1 - 9 {x}^{2}}}$

#### Explanation:

We will use the quotient rule:

$\frac{d}{\mathrm{dx}} \left[f \frac{x}{g} \left(x\right)\right] = y ' = \frac{f ' \left(x\right) \cdot g \left(x\right) - g ' \left(x\right) \cdot f \left(x\right)}{g \left(x\right)} ^ 2$

But before that however, let's find the derivative of $\arcsin \left(3 x\right)$

$- - - - - - - - - - - - - - - - - - - -$

Let $y = \arcsin \left(3 x\right)$

Take the sines of both sides

$\sin \left(y\right) = 3 x$

Differentiate both sides W.R.T $x$

$\frac{\mathrm{dy}}{\mathrm{dx}} \cdot \cos \left(y\right) = 3$

Divide $\cos \left(y\right)$ to both sides

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3}{\cos} \left(y\right)$

We now must rewrite in terms of $x$

Since , color(blue)(sin(y)=(3x)/1

Then, color(red)(cos(y)=sqrt(1-9x^2)/1=sqrt(1-9x^2)

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3}{\sqrt{1 - 9 {x}^{2}}}$

$- - - - - - - - - - - - - - - - - - - -$

Now finding the derivative:

Let $f \left(x\right) = \arcsin \left(3 x\right)$ & $g \left(x\right) = x$

So $f ' \left(x\right) = \frac{3}{\sqrt{1 - 9 {x}^{2}}}$ & $g ' \left(x\right) = 1$

Substituting into the quotient rule we get:

$y ' = \frac{\frac{3}{\sqrt{1 - 9 {x}^{2}}} \cdot x - 1 \cdot \arcsin \left(3 x\right)}{x} ^ 2$

Simplify:

$y ' = \frac{\frac{3 x}{\sqrt{1 - 9 {x}^{2}}} - \arcsin \left(3 x\right)}{x} ^ 2$

Combine Fractions in the numerator:

color(blue)((3x)/sqrt(1-9x^2)-arcsin(3x)*(sqrt(1-9x^2)/sqrt(1-9x^2))

$\textcolor{b l u e}{\frac{3 x}{\sqrt{1 - 9 {x}^{2}}} - \frac{\arcsin \left(3 x\right) \sqrt{1 - 9 {x}^{2}}}{\sqrt{1 - 9 {x}^{2}}}}$

$y ' = \frac{\frac{3 x - \arcsin \left(3 x\right) \sqrt{1 - 9 {x}^{2}}}{\sqrt{1 - 9 {x}^{2}}}}{x} ^ 2$

Apply the fraction rule for further simplification

$y ' = \frac{3 x - \arcsin \left(3 x\right) \sqrt{1 - 9 {x}^{2}}}{\sqrt{1 - 9 {x}^{2}}} \cdot \frac{1}{x} ^ 2$

$y ' = \frac{3 x - \arcsin \left(3 x\right) \sqrt{1 - 9 {x}^{2}}}{{x}^{2} \sqrt{1 - 9 {x}^{2}}}$