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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $x$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Combine terms.

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Step 5.2.1

Add $0$ and $0$ .

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

Step 5.2.2

Add $0$ and $x$ .

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

Step 5.3

Reorder terms.

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