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

\frac{d}{dx}\left(\left(\frac{x}{2}\right)^{\frac{2}{3}}\right)

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Answer

$\frac{\sqrt[4]{6}}{3}x^{-\frac{1}{3}}$

Step by step solution

Problem

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

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

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

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

The derivative of the constant function is equal to zero

$\frac{2}{3}\left(\frac{x}{2}\right)^{-\frac{1}{3}}\cdot\frac{0\left(-1\right)x+2\frac{d}{dx}\left(x\right)}{4}$
4

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

$\frac{2}{3}\left(\frac{x}{2}\right)^{-\frac{1}{3}}\cdot\frac{0+2\frac{d}{dx}\left(x\right)}{4}$
5

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

$\frac{0+2\cdot 1}{4}\cdot \frac{2}{3}\left(\frac{x}{2}\right)^{-\frac{1}{3}}$
6

Multiply $1$ times $2$

$\frac{0+2}{4}\cdot \frac{2}{3}\left(\frac{x}{2}\right)^{-\frac{1}{3}}$
7

Add the values $2$ and $0$

$\frac{2}{4}\cdot \frac{2}{3}\left(\frac{x}{2}\right)^{-\frac{1}{3}}$
8

Divide $2$ by $4$

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

Multiply $\frac{2}{3}$ times $\frac{1}{2}$

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

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

$\frac{1}{3}\cdot\frac{x^{-\frac{1}{3}}}{\frac{\sqrt[3]{6}}{2}}$
11

Simplify the fraction

$\frac{\sqrt[4]{6}}{3}x^{-\frac{1}{3}}$
12

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

$\frac{\sqrt[4]{6}}{3}\cdot\frac{1}{\sqrt[3]{x}}$
13

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=\frac{\sqrt[4]{6}}{3}$ and $x=\sqrt[3]{x}$

$\frac{\frac{\sqrt[4]{6}}{3}}{\sqrt[3]{x}}$
14

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

$\frac{\sqrt[4]{6}}{3}x^{-\frac{1}{3}}$

Answer

$\frac{\sqrt[4]{6}}{3}x^{-\frac{1}{3}}$

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.24 seconds

Views:

76