Calculus Examples

$7 \sinh (\ln (t))$

Step 1

Since $7$ is constant with respect to $t$ , the derivative of $7 \sinh (\ln (t))$ with respect to $t$ is $7 \frac{d}{d t} [\sinh (\ln (t))]$ .

$7 \frac{d}{d t} [\sinh (\ln (t))]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \sinh (t)$ and $g (t) = \ln (t)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\ln (t)$ .

$7 (\frac{d}{d u} [\sinh (u)] \frac{d}{d t} [\ln (t)])$

Step 2.2

The derivative of $\sinh (u)$ with respect to $u$ is $\cosh (u)$ .

$7 (\cosh (u) \frac{d}{d t} [\ln (t)])$

Step 2.3

Replace all occurrences of $u$ with $\ln (t)$ .

$7 (\cosh (\ln (t)) \frac{d}{d t} [\ln (t)])$

Step 3

The derivative of $\ln (t)$ with respect to $t$ is $\frac{1}{t}$ .

$7 \cosh (\ln (t)) \frac{1}{t}$

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{t}$ and $7$ .

$\frac{7}{t} \cosh (\ln (t))$

Step 4.2

Combine $\frac{7}{t}$ and $\cosh (\ln (t))$ .

$\frac{7 \cosh (\ln (t))}{t}$