Calculus Examples

$f (x) = x^{2} \cot (x) - \frac{1}{x^{2}}$

Step 1

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

$\frac{d}{d x} [x^{2} \cot (x)] + \frac{d}{d x} [- \frac{1}{x^{2}}]$

Step 2

Evaluate $\frac{d}{d x} [x^{2} \cot (x)]$ .

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = \cot (x)$ .

$x^{2} \frac{d}{d x} [\cot (x)] + \cot (x) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{1}{x^{2}}]$

Step 2.2

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{1}{x^{2}}]$

Step 2.3

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

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) + \frac{d}{d x} [- \frac{1}{x^{2}}]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{1}{x^{2}}]$ .

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{2}}$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 3.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 3.3.2

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

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - (- u^{- 2} \frac{d}{d x} [x^{2}]) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 3.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 3.4

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

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - (- {(x^{2})}^{- 2} (2 x)) + \frac{1}{x^{2}} \frac{d}{d x} [- 1]$

Step 3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - (- {(x^{2})}^{- 2} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 3.6

Multiply the exponents in ${(x^{2})}^{- 2}$ .

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Step 3.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - (- x^{2 \cdot - 2} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 3.6.2

Multiply $2$ by $- 2$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - (- x^{- 4} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 3.7

Multiply $2$ by $- 1$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - (- 2 x^{- 4} x) + \frac{1}{x^{2}} \cdot 0$

Step 3.8

Raise $x$ to the power of $1$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - (- 2 (x^{1} x^{- 4})) + \frac{1}{x^{2}} \cdot 0$

Step 3.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - (- 2 x^{1 - 4}) + \frac{1}{x^{2}} \cdot 0$

Step 3.10

Subtract $4$ from $1$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) - (- 2 x^{- 3}) + \frac{1}{x^{2}} \cdot 0$

Step 3.11

Multiply $- 2$ by $- 1$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) + 2 x^{- 3} + \frac{1}{x^{2}} \cdot 0$

Step 3.12

Multiply $\frac{1}{x^{2}}$ by $0$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) + 2 x^{- 3} + 0$

Step 3.13

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

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) + 2 x^{- 3}$

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Combine $2$ and $\frac{1}{x^{3}}$ .

$x^{2} (- \csc^{2} (x)) + \cot (x) (2 x) + \frac{2}{x^{3}}$

Step 4.3

Reorder terms.

$- x^{2} \csc^{2} (x) + 2 x \cot (x) + \frac{2}{x^{3}}$