Derive the function arcsin(x/a) with respect to x

\frac{d}{dx}\left(arcsin\left(\frac{x}{a}\right)\right)

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Answer

$\frac{1}{a\sqrt{1-x^2a^{-2}}}$

Step by step solution

Problem

$\frac{d}{dx}\left(arcsin\left(\frac{x}{a}\right)\right)$
1

Taking the derivative of arcsine

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

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

The derivative of the constant function is equal to zero

$\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}\cdot\frac{0\left(-1\right)x+a\frac{d}{dx}\left(x\right)}{a^2}$
4

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

$\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}\cdot\frac{0+a\frac{d}{dx}\left(x\right)}{a^2}$
5

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

$\frac{0+1a}{a^2}\cdot\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}$
6

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

$\frac{a}{a^2}\cdot\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}$
7

Simplifying the fraction by $a$

$\frac{1}{a}\cdot\frac{1}{\sqrt{1-\left(\frac{x}{a}\right)^2}}$
8

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

$\frac{1}{a}\cdot\frac{1}{\sqrt{1-\frac{x^2}{a^2}}}$
9

Multiplying fractions

$\frac{1}{a\sqrt{1-\frac{x^2}{a^2}}}$
10

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

$\frac{1}{a\sqrt{1-a^{-2}x^2}}$
11

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

$\frac{1}{a\sqrt{1-\left(\frac{1}{a^{2}}\right)x^2}}$
12

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

$\frac{1}{a\sqrt{1-1\cdot 1x^2a^{-2}}}$
13

Multiply $1$ times $-1$

$\frac{1}{a\sqrt{1-x^2a^{-2}}}$

Answer

$\frac{1}{a\sqrt{1-x^2a^{-2}}}$

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.41 seconds

Views:

113