Calculus Examples

$y = \frac{1}{4} \cdot \sinh (2 x) - \frac{x}{2}$

Step 1

By the Sum Rule, the derivative of $\frac{1}{4} \cdot \sinh (2 x) - \frac{x}{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4} \cdot \sinh (2 x)] + \frac{d}{d x} [- \frac{x}{2}]$ .

$\frac{d}{d x} [\frac{1}{4} \cdot \sinh (2 x)] + \frac{d}{d x} [- \frac{x}{2}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{4} \cdot \sinh (2 x)]$ .

Tap for more steps...

Step 2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} \cdot \sinh (2 x)$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\sinh (2 x)]$ .

$\frac{1}{4} \frac{d}{d x} [\sinh (2 x)] + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.2

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

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$\frac{1}{4} (\frac{d}{d u} [\sinh (u)] \frac{d}{d x} [2 x]) + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.2.2

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

$\frac{1}{4} (\cosh (u) \frac{d}{d x} [2 x]) + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.2.3

Replace all occurrences of $u$ with $2 x$ .

$\frac{1}{4} (\cosh (2 x) \frac{d}{d x} [2 x]) + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.3

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

$\frac{1}{4} (\cosh (2 x) (2 \frac{d}{d x} [x])) + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.4

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

$\frac{1}{4} (\cosh (2 x) (2 \cdot 1)) + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.5

Multiply $2$ by $1$ .

$\frac{1}{4} (\cosh (2 x) \cdot 2) + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.6

Move $2$ to the left of $\cosh (2 x)$ .

$\frac{1}{4} (2 \cdot \cosh (2 x)) + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.7

Combine $2$ and $\frac{1}{4}$ .

$\frac{2}{4} \cosh (2 x) + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.8

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

$\frac{2 \cosh (2 x)}{4} + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.9

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

Tap for more steps...

Step 2.9.1

Factor $2$ out of $2 \cosh (2 x)$ .

$\frac{2 (\cosh (2 x))}{4} + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.9.2

Cancel the common factors.

Tap for more steps...

Step 2.9.2.1

Factor $2$ out of $4$ .

$\frac{2 \cosh (2 x)}{2 \cdot 2} + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.9.2.2

Cancel the common factor.

$\frac{2 \cosh (2 x)}{2 \cdot 2} + \frac{d}{d x} [- \frac{x}{2}]$

Step 2.9.2.3

Rewrite the expression.

$\frac{\cosh (2 x)}{2} + \frac{d}{d x} [- \frac{x}{2}]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{x}{2}]$ .

Tap for more steps...

Step 3.1

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

$\frac{\cosh (2 x)}{2} - \frac{1}{2} \frac{d}{d x} [x]$

Step 3.2

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

$\frac{\cosh (2 x)}{2} - \frac{1}{2} \cdot 1$

Step 3.3

Multiply $- 1$ by $1$ .

$\frac{\cosh (2 x)}{2} - \frac{1}{2}$