Calculus Examples

$y = sech (x) \tanh (x)$

Step 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 $f (x) = sech (x)$ and $g (x) = \tanh (x)$ .

$sech (x) \frac{d}{d x} [\tanh (x)] + \tanh (x) \frac{d}{d x} [sech (x)]$

Step 2

The derivative of $\tanh (x)$ with respect to $x$ is ${sech}^{2} (x)$ .

$sech (x) {sech}^{2} (x) + \tanh (x) \frac{d}{d x} [sech (x)]$

Step 3

Multiply $sech (x)$ by ${sech}^{2} (x)$ by adding the exponents.

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Step 3.1

Multiply $sech (x)$ by ${sech}^{2} (x)$ .

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Step 3.1.1

Raise $sech (x)$ to the power of $1$ .

${sech}^{1} (x) {sech}^{2} (x) + \tanh (x) \frac{d}{d x} [sech (x)]$

Step 3.1.2

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

${sech (x)}^{1 + 2} + \tanh (x) \frac{d}{d x} [sech (x)]$

Step 3.2

Add $1$ and $2$ .

${sech}^{3} (x) + \tanh (x) \frac{d}{d x} [sech (x)]$

Step 4

The derivative of $sech (x)$ with respect to $x$ is $- sech (x) \tanh (x)$ .

${sech}^{3} (x) + \tanh (x) (- sech (x) \tanh (x))$

Step 5

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

${sech}^{3} (x) - sech (x) (\tanh^{1} (x) \tanh (x))$

Step 6

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

${sech}^{3} (x) - sech (x) (\tanh^{1} (x) \tanh^{1} (x))$

Step 7

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

${sech}^{3} (x) - sech (x) {\tanh (x)}^{1 + 1}$

Step 8

Add $1$ and $1$ .

${sech}^{3} (x) - sech (x) \tanh^{2} (x)$

Step 9

Reorder terms.

${sech}^{3} (x) - \tanh^{2} (x) sech (x)$