Derive the function arcsin(x/a)+((a^2-1x^2)^0.5)/x with respect to x

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

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Answer

$\frac{a\left(-\frac{1}{\sqrt{a^2-x^2}}x^2-\sqrt{a^2-x^2}\right)\sqrt{\frac{-x^2}{a^2}+1}+x^2}{ax^2\sqrt{\frac{-x^2}{a^2}+1}}$

Step by step solution

Problem

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

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

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

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

The derivative of the linear function is equal to $1$

$\frac{1\left(-1\right)\sqrt{a^2-x^2}+x\frac{d}{dx}\left(\sqrt{a^2-x^2}\right)}{x^2}+\frac{d}{dx}\left(arcsin\left(\frac{x}{a}\right)\right)$
4

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

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

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

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

The derivative of the constant function is equal to zero

$\frac{1\left(-1\right)\sqrt{a^2-x^2}+\frac{1}{2}x\left(a^2-x^2\right)^{-\frac{1}{2}}\left(\frac{d}{dx}\left(-x^2\right)+0\right)}{x^2}+\frac{d}{dx}\left(arcsin\left(\frac{x}{a}\right)\right)$
7

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{1\left(-1\right)\sqrt{a^2-x^2}+\frac{1}{2}x\left(a^2-x^2\right)^{-\frac{1}{2}}\left(0-\frac{d}{dx}\left(x^2\right)\right)}{x^2}+\frac{d}{dx}\left(arcsin\left(\frac{x}{a}\right)\right)$
8

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

$\frac{1\left(-1\right)\sqrt{a^2-x^2}+\frac{1}{2}x\left(0-1\cdot 2x\right)\left(a^2-x^2\right)^{-\frac{1}{2}}}{x^2}+\frac{d}{dx}\left(arcsin\left(\frac{x}{a}\right)\right)$
9

Taking the derivative of arcsine

$\frac{1\left(-1\right)\sqrt{a^2-x^2}+\frac{1}{2}x\left(0-1\cdot 2x\right)\left(a^2-x^2\right)^{-\frac{1}{2}}}{x^2}+\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}\cdot\frac{d}{dx}\left(\frac{x}{a}\right)$
10

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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{1\left(-1\right)\sqrt{a^2-x^2}+\frac{1}{2}x\left(0-1\cdot 2x\right)\left(a^2-x^2\right)^{-\frac{1}{2}}}{x^2}+\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}\cdot\frac{a\frac{d}{dx}\left(x\right)-x\frac{d}{dx}\left(a\right)}{a^2}$
11

The derivative of the constant function is equal to zero

$\frac{1\left(-1\right)\sqrt{a^2-x^2}+\frac{1}{2}x\left(0-1\cdot 2x\right)\left(a^2-x^2\right)^{-\frac{1}{2}}}{x^2}+\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}\cdot\frac{0\left(-1\right)x+a\frac{d}{dx}\left(x\right)}{a^2}$
12

Any expression multiplied by $0$ is equal to $0$

$\frac{1\left(-1\right)\sqrt{a^2-x^2}+\frac{1}{2}x\left(0-1\cdot 2x\right)\left(a^2-x^2\right)^{-\frac{1}{2}}}{x^2}+\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}\cdot\frac{0+a\frac{d}{dx}\left(x\right)}{a^2}$
13

The derivative of the linear function is equal to $1$

$\frac{1\left(-1\right)\sqrt{a^2-x^2}+\frac{1}{2}x\left(0-1\cdot 2x\right)\left(a^2-x^2\right)^{-\frac{1}{2}}}{x^2}+\frac{0+1a}{a^2}\cdot\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}$
14

Multiply $2$ times $-1$

$\frac{\frac{1}{2}x\left(0-2x\right)\left(a^2-x^2\right)^{-\frac{1}{2}}-\sqrt{a^2-x^2}}{x^2}+\frac{0+1a}{a^2}\cdot\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}$
15

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

$\frac{\frac{1}{2}\left(-2\right)x\cdot x\left(a^2-x^2\right)^{-\frac{1}{2}}-\sqrt{a^2-x^2}}{x^2}+\frac{a}{a^2}\cdot\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}$
16

Multiply $-2$ times $\frac{1}{2}$

$\frac{-x\cdot x\left(a^2-x^2\right)^{-\frac{1}{2}}-\sqrt{a^2-x^2}}{x^2}+\frac{a}{a^2}\cdot\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}$
17

When multiplying exponents with same base you can add the exponents

$\frac{-\left(a^2-x^2\right)^{-\frac{1}{2}}x^2-\sqrt{a^2-x^2}}{x^2}+\frac{a}{a^2}\cdot\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}$
18

Simplifying the fraction by $a$

$\frac{-\left(a^2-x^2\right)^{-\frac{1}{2}}x^2-\sqrt{a^2-x^2}}{x^2}+\frac{1}{a}\cdot\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}$
19

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

$\frac{-\left(a^2-x^2\right)^{-\frac{1}{2}}x^2-\sqrt{a^2-x^2}}{x^2}+\frac{1}{a}\cdot\frac{1}{\sqrt{1-\frac{x^2}{a^2}}}$
20

Multiplying fractions

$\frac{-\left(a^2-x^2\right)^{-\frac{1}{2}}x^2-\sqrt{a^2-x^2}}{x^2}+\frac{1}{a\sqrt{1-\frac{x^2}{a^2}}}$
21

Unifying fractions

$\frac{a\left(-\left(a^2-x^2\right)^{-\frac{1}{2}}x^2-\sqrt{a^2-x^2}\right)\sqrt{\frac{-x^2}{a^2}+1}+x^2}{ax^2\sqrt{\frac{-x^2}{a^2}+1}}$
22

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

$\frac{a\left(-\frac{1}{\sqrt{a^2-x^2}}x^2-\sqrt{a^2-x^2}\right)\sqrt{\frac{-x^2}{a^2}+1}+x^2}{ax^2\sqrt{\frac{-x^2}{a^2}+1}}$

Answer

$\frac{a\left(-\frac{1}{\sqrt{a^2-x^2}}x^2-\sqrt{a^2-x^2}\right)\sqrt{\frac{-x^2}{a^2}+1}+x^2}{ax^2\sqrt{\frac{-x^2}{a^2}+1}}$

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