Calculus Examples

$\csc (x)$

Find the first derivative.

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

$f' (x) = - \csc (x) \cot (x)$

Reorder the factors of $- \csc (x) \cot (x)$ .

$f' (x) = - \cot (x) \csc (x)$

Find the second derivative.

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Since $- 1$ is constant with respect to $x$ , the derivative of $- \cot (x) \csc (x)$ with respect to $x$ is $- \frac{d}{d x} [\cot (x) \csc (x)]$ .

$f'' (x) = - \frac{d}{d x} (\cot (x) \csc (x))$

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 $f (x) = \cot (x)$ and $g (x) = \csc (x)$ .

$f'' (x) = - (\cot (x) \frac{d}{d x} (\csc (x)) + \csc (x) \frac{d}{d x} (\cot (x)))$

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

$f'' (x) = - (\cot (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} (\cot (x)))$

Raise $\cot (x)$ to the power of $1$ .

$f'' (x) = - (- \csc (x) (\cot (x) \cot (x)) + \csc (x) \frac{d}{d x} (\cot (x)))$

Raise $\cot (x)$ to the power of $1$ .

$f'' (x) = - (- \csc (x) (\cot (x) \cot (x)) + \csc (x) \frac{d}{d x} (\cot (x)))$

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

$f'' (x) = - (- \csc (x) {\cot (x)}^{1 + 1} + \csc (x) \frac{d}{d x} (\cot (x)))$

Add $1$ and $1$ .

$f'' (x) = - (- \csc (x) \cot^{2} (x) + \csc (x) \frac{d}{d x} (\cot (x)))$

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

$f'' (x) = - (- \csc (x) \cot^{2} (x) + \csc (x) (- \csc^{2} (x)))$

Raise $\csc (x)$ to the power of $1$ .

$f'' (x) = - (- \csc (x) \cot^{2} (x) - (\csc (x) \csc^{2} (x)))$

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

$f'' (x) = - (- \csc (x) \cot^{2} (x) - {\csc (x)}^{1 + 2})$

Add $1$ and $2$ .

$f'' (x) = - (- \csc (x) \cot^{2} (x) - \csc^{3} (x))$

Simplify.

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Apply the distributive property.

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

Combine terms.

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Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\csc (x) \cot^{2} (x)) + \csc^{3} (x)$

Multiply $\csc (x)$ by $1$ .

$f'' (x) = \csc (x) \cot^{2} (x) + \csc^{3} (x)$

Multiply $- 1$ by $- 1$ .

$f'' (x) = \csc (x) \cot^{2} (x) + 1 \csc^{3} (x)$

Multiply $\csc^{3} (x)$ by $1$ .

$f'' (x) = \csc (x) \cot^{2} (x) + \csc^{3} (x)$

Reorder terms.

$f'' (x) = \cot^{2} (x) \csc (x) + \csc^{3} (x)$