# Step-by-step Solution

## Derive the function $arcsin\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)$ with respect to x

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### Videos

$\frac{1}{\sqrt{1-\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)^2}}\cdot\frac{\frac{1}{2}x^{-\frac{1}{2}}\sin\left(x^2\right)-2x\left(\sqrt{x}+3\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2}$

## Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(arcsin\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)\right)$
1

Taking the derivative of arcsine

$\frac{1}{\sqrt{1-\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)^2}}\cdot\frac{d}{dx}\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)$
2

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{1}{\sqrt{1-\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)^2}}\cdot\frac{\frac{d}{dx}\left(\sqrt{x}+3\right)\sin\left(x^2\right)-\left(\sqrt{x}+3\right)\frac{d}{dx}\left(\sin\left(x^2\right)\right)}{\sin\left(x^2\right)^2}$

$\frac{1}{\sqrt{1-\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)^2}}\cdot\frac{\frac{1}{2}x^{-\frac{1}{2}}\sin\left(x^2\right)-2x\left(\sqrt{x}+3\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2}$
$\frac{d}{dx}\left(arcsin\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)\right)$

### Main topic:

Differential calculus

~ 0.61 seconds