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

Derive the function arcsin((x^0.5+3)/(sin(x^2))) with respect to x

Answer

$\frac{\frac{1}{2}\sin\left(x^2\right)-2\sqrt{x^{3}}\left(3+\sqrt{x}\right)\cos\left(x^2\right)}{\sqrt{x}\sin\left(x^2\right)^2\sqrt{\frac{-\left(6\sqrt{x}+x\right)-9}{\sin\left(x^2\right)^2}+1}}$

Step-by-step explanation

Problem

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

Taking the derivative of arcsine

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

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Answer

$\frac{\frac{1}{2}\sin\left(x^2\right)-2\sqrt{x^{3}}\left(3+\sqrt{x}\right)\cos\left(x^2\right)}{\sqrt{x}\sin\left(x^2\right)^2\sqrt{\frac{-\left(6\sqrt{x}+x\right)-9}{\sin\left(x^2\right)^2}+1}}$
$\frac{d}{dx}\left(arcsin\left(\frac{\sqrt{x}+3}{\sin\left(x^2\right)}\right)\right)$

Main topic:

Differential calculus

Used formulas:

4. See formulas

Time to solve it:

~ 1.58 seconds