Calculus Examples

$x^{3} + y^{3} + \frac{3}{x y}$

Step 1

Differentiate.

Tap for more steps...

Step 1.1

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

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [y^{3}] + \frac{d}{d x} [\frac{3}{x y}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [y^{3}] + \frac{d}{d x} [\frac{3}{x y}]$

Step 1.3

Since $y^{3}$ is constant with respect to $x$ , the derivative of $y^{3}$ with respect to $x$ is $0$ .

$3 x^{2} + 0 + \frac{d}{d x} [\frac{3}{x y}]$

Step 2

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

Tap for more steps...

Step 2.1

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

$3 x^{2} + 0 + \frac{3}{y} \frac{d}{d x} [\frac{1}{x}]$

Step 2.2

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

$3 x^{2} + 0 + \frac{3}{y} \frac{d}{d x} [x^{- 1}]$

Step 2.3

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

$3 x^{2} + 0 + \frac{3}{y} (- x^{- 2})$

Step 2.4

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

$3 x^{2} + 0 - \frac{3 x^{- 2}}{y}$

Step 2.5

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

$3 x^{2} + 0 - \frac{3}{y x^{2}}$

Step 3

Add $3 x^{2}$ and $0$ .

$3 x^{2} - \frac{3}{y x^{2}}$