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

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

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Answer

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

Step by step solution

Problem

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

Taking the derivative of arcsine

$\frac{d}{dx}\left(\frac{3+\sqrt{x}}{\sin\left(x^2\right)}\right)\frac{1}{\sqrt{1-\left(\frac{3+\sqrt{x}}{\sin\left(x^2\right)}\right)^2}}$
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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\frac{\sin\left(x^2\right)\frac{d}{dx}\left(3+\sqrt{x}\right)-\left(3+\sqrt{x}\right)\frac{d}{dx}\left(x^2\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2}\cdot\frac{1}{\sqrt{1-\left(\frac{3+\sqrt{x}}{\sin\left(x^2\right)}\right)^2}}$
4

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{\sin\left(x^2\right)\frac{d}{dx}\left(3+\sqrt{x}\right)-1\cdot 2x\left(3+\sqrt{x}\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2}\cdot\frac{1}{\sqrt{1-\left(\frac{3+\sqrt{x}}{\sin\left(x^2\right)}\right)^2}}$
5

The derivative of a sum of two functions is the sum of the derivatives of each function

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

The derivative of the constant function is equal to zero

$\frac{\sin\left(x^2\right)\left(0+\frac{d}{dx}\left(\sqrt{x}\right)\right)-1\cdot 2x\left(3+\sqrt{x}\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2}\cdot\frac{1}{\sqrt{1-\left(\frac{3+\sqrt{x}}{\sin\left(x^2\right)}\right)^2}}$
7

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{\left(0+\frac{1}{2}x^{-\frac{1}{2}}\right)\sin\left(x^2\right)-1\cdot 2x\left(3+\sqrt{x}\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2}\cdot\frac{1}{\sqrt{1-\left(\frac{3+\sqrt{x}}{\sin\left(x^2\right)}\right)^2}}$
8

Multiply $2$ times $-1$

$\frac{\left(0+\frac{1}{2}x^{-\frac{1}{2}}\right)\sin\left(x^2\right)-2x\left(3+\sqrt{x}\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2}\cdot\frac{1}{\sqrt{1-\left(\frac{3+\sqrt{x}}{\sin\left(x^2\right)}\right)^2}}$
9

$x+0=x$, where $x$ is any expression

$\frac{\frac{1}{2}x^{-\frac{1}{2}}\sin\left(x^2\right)-2x\left(3+\sqrt{x}\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2}\cdot\frac{1}{\sqrt{1-\left(\frac{3+\sqrt{x}}{\sin\left(x^2\right)}\right)^2}}$
10

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$\frac{\frac{1}{2}x^{-\frac{1}{2}}\sin\left(x^2\right)-2x\left(3+\sqrt{x}\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2}\cdot\frac{1}{\sqrt{1-\frac{\left(3+\sqrt{x}\right)^2}{\sin\left(x^2\right)^2}}}$
11

Multiplying fractions

$\frac{\frac{1}{2}x^{-\frac{1}{2}}\sin\left(x^2\right)-2x\left(3+\sqrt{x}\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2\sqrt{1-\frac{\left(3+\sqrt{x}\right)^2}{\sin\left(x^2\right)^2}}}$
12

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{\frac{1}{2}\cdot\frac{1}{\sqrt{x}}\sin\left(x^2\right)-2x\left(3+\sqrt{x}\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2\sqrt{1-\frac{\left(3+\sqrt{x}\right)^2}{\sin\left(x^2\right)^2}}}$
13

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=\frac{1}{2}$ and $x=\sqrt{x}$

$\frac{\frac{\frac{1}{2}}{\sqrt{x}}\sin\left(x^2\right)-2x\left(3+\sqrt{x}\right)\cos\left(x^2\right)}{\sin\left(x^2\right)^2\sqrt{1-\frac{\left(3+\sqrt{x}\right)^2}{\sin\left(x^2\right)^2}}}$
14

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

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

Answer

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

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

1.22 seconds

Views:

102