Calculus Examples

$y = x + \frac{4}{3 x}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x + \frac{4}{3 x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{4}{3 x}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{4}{3 x}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [\frac{4}{3 x}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{4}{3 x}]$ .

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Step 2.1

Since $\frac{4}{3}$ is constant with respect to $x$ , the derivative of $\frac{4}{3 x}$ with respect to $x$ is $\frac{4}{3} \frac{d}{d x} [\frac{1}{x}]$ .

$1 + \frac{4}{3} \frac{d}{d x} [\frac{1}{x}]$

Step 2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$1 + \frac{4}{3} \frac{d}{d x} [x^{- 1}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 1$ .

$1 + \frac{4}{3} (- x^{- 2})$

Step 2.4

Combine $\frac{4}{3}$ and $x^{- 2}$ .

$1 - \frac{4 x^{- 2}}{3}$

Step 2.5

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 - \frac{4}{3 x^{2}}$

Step 3

Reorder terms.

$- \frac{4}{3 x^{2}} + 1$