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Step-by-step Solution

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

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Answer

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

Step-by-step explanation

Problem to solve:

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

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

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Answer

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

Main topic:

Differential calculus

Used formulas:

3. See formulas

Time to solve it:

~ 0.8 seconds

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