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

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

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Answer

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

Step by step solution

Problem

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

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

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

Taking the derivative of arccosine

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

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

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

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

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

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

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

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

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

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

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

Multiply $3$ times $1$

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

Applying the power of a power property

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

Multiplying the fraction and term

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

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

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

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=-\frac{1}{3}$ and $x=\sqrt{3x+x^3}$

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

Multiplying the fraction and term

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

Simplifying the fraction

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

Answer

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

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.54 seconds

Views:

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