Calculus Examples

$4 \ln (\tanh (\frac{x}{2}))$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \ln (\tanh (\frac{x}{2}))$ with respect to $x$ is $4 \frac{d}{d x} [\ln (\tanh (\frac{x}{2}))]$ .

$4 \frac{d}{d x} [\ln (\tanh (\frac{x}{2}))]$

Step 2

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

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\tanh (\frac{x}{2})$ .

$4 (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\tanh (\frac{x}{2})])$

Step 2.2

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

$4 (\frac{1}{u_{1}} \frac{d}{d x} [\tanh (\frac{x}{2})])$

Step 2.3

Replace all occurrences of $u_{1}$ with $\tanh (\frac{x}{2})$ .

$4 (\frac{1}{\tanh (\frac{x}{2})} \frac{d}{d x} [\tanh (\frac{x}{2})])$

Step 3

Combine $\frac{1}{\tanh (\frac{x}{2})}$ and $4$ .

$\frac{4}{\tanh (\frac{x}{2})} \frac{d}{d x} [\tanh (\frac{x}{2})]$

Step 4

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

Tap for more steps...

Step 4.1

To apply the Chain Rule, set $u_{2}$ as $\frac{x}{2}$ .

$\frac{4}{\tanh (\frac{x}{2})} (\frac{d}{d u_{2}} [\tanh (u_{2})] \frac{d}{d x} [\frac{x}{2}])$

Step 4.2

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

$\frac{4}{\tanh (\frac{x}{2})} ({sech}^{2} (u_{2}) \frac{d}{d x} [\frac{x}{2}])$

Step 4.3

Replace all occurrences of $u_{2}$ with $\frac{x}{2}$ .

$\frac{4}{\tanh (\frac{x}{2})} ({sech}^{2} (\frac{x}{2}) \frac{d}{d x} [\frac{x}{2}])$

Step 5

Differentiate.

Tap for more steps...

Step 5.1

Combine ${sech}^{2} (\frac{x}{2})$ and $\frac{4}{\tanh (\frac{x}{2})}$ .

$\frac{{sech}^{2} (\frac{x}{2}) \cdot 4}{\tanh (\frac{x}{2})} \frac{d}{d x} [\frac{x}{2}]$

Step 5.2

Move $4$ to the left of ${sech}^{2} (\frac{x}{2})$ .

$\frac{4 \cdot {sech}^{2} (\frac{x}{2})}{\tanh (\frac{x}{2})} \frac{d}{d x} [\frac{x}{2}]$

Step 5.3

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

$\frac{4 {sech}^{2} (\frac{x}{2})}{\tanh (\frac{x}{2})} (\frac{1}{2} \frac{d}{d x} [x])$

Step 5.4

Simplify terms.

Tap for more steps...

Step 5.4.1

Multiply $\frac{1}{2}$ by $\frac{4 {sech}^{2} (\frac{x}{2})}{\tanh (\frac{x}{2})}$ .

$\frac{4 {sech}^{2} (\frac{x}{2})}{2 \tanh (\frac{x}{2})} \frac{d}{d x} [x]$

Step 5.4.2

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 5.4.2.1

Factor $2$ out of $4 {sech}^{2} (\frac{x}{2})$ .

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

Step 5.4.2.2

Cancel the common factors.

Tap for more steps...

Step 5.4.2.2.1

Factor $2$ out of $2 \tanh (\frac{x}{2})$ .

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

Step 5.4.2.2.2

Cancel the common factor.

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

Step 5.4.2.2.3

Rewrite the expression.

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

Step 5.5

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

$\frac{2 {sech}^{2} (\frac{x}{2})}{\tanh (\frac{x}{2})} \cdot 1$

Step 5.6

Multiply $\frac{2 {sech}^{2} (\frac{x}{2})}{\tanh (\frac{x}{2})}$ by $1$ .

$\frac{2 {sech}^{2} (\frac{x}{2})}{\tanh (\frac{x}{2})}$