Calculus Examples

$x^{2} + \frac{1}{x} + x y + \frac{1}{y} + y^{2}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{2} + \frac{1}{x} + x y + \frac{1}{y} + y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [x y] + \frac{d}{d x} [\frac{1}{y}] + \frac{d}{d x} [y^{2}]$ .

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $y$ by $1$ .

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

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $\frac{1}{y}$ is constant with respect to $x$ , the derivative of $\frac{1}{y}$ with respect to $x$ is $0$ .

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

Step 4.2

Since $y^{2}$ is constant with respect to $x$ , the derivative of $y^{2}$ with respect to $x$ is $0$ .

$2 x - x^{- 2} + y + 0 + 0$

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 x - \frac{1}{x^{2}} + y + 0 + 0$

Step 5.2

Combine terms.

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Step 5.2.1

Add $2 x - \frac{1}{x^{2}} + y$ and $0$ .

$2 x - \frac{1}{x^{2}} + y + 0$

Step 5.2.2

Add $2 x - \frac{1}{x^{2}} + y$ and $0$ .

$2 x - \frac{1}{x^{2}} + y$

Step 5.3

Reorder terms.

$2 x + y - \frac{1}{x^{2}}$