Calculus Examples

$f (x) = \cot (x)$

Step 1

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

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

Step 2

Find the second derivative.

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Since $- 1$ is constant with respect to $x$ , the derivative of $- \csc^{2} (x)$ with respect to $x$ is $- \frac{d}{d x} [\csc^{2} (x)]$ .

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

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

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To apply the Chain Rule, set $u$ as $\csc (x)$ .

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

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

$f'' (x) = - (2 u \frac{d}{d x} (\csc (x)))$

Replace all occurrences of $u$ with $\csc (x)$ .

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

Multiply $2$ by $- 1$ .

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

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

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

Multiply $- 1$ by $- 2$ .

$f'' (x) = 2 \csc (x) (\csc (x) \cot (x))$

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

$f'' (x) = 2 (\csc (x) \csc (x)) \cot (x)$

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

$f'' (x) = 2 (\csc (x) \csc (x)) \cot (x)$

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

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

Add $1$ and $1$ .

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

Step 3

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

$2 \csc^{2} (x) \cot (x)$