Calculus Examples

$y = 2 \tan (\frac{1}{2} x) - x$

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [2 \tan (\frac{1}{2} x)]$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$2 (\sec^{2} (u) \frac{d}{d x} [\frac{x}{2}]) + \frac{d}{d x} [- x]$

Step 2.3.3

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

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

Step 2.4

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

$2 (\sec^{2} (\frac{x}{2}) (\frac{1}{2} \frac{d}{d x} [x])) + \frac{d}{d x} [- x]$

Step 2.5

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

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

Step 2.6

Multiply $\frac{1}{2}$ by $1$ .

$2 (\sec^{2} (\frac{x}{2}) \frac{1}{2}) + \frac{d}{d x} [- x]$

Step 2.7

Combine $\sec^{2} (\frac{x}{2})$ and $\frac{1}{2}$ .

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

Step 2.8

Combine $2$ and $\frac{\sec^{2} (\frac{x}{2})}{2}$ .

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

Step 2.9

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.9.1

Cancel the common factor.

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

Step 2.9.2

Divide $\sec^{2} (\frac{x}{2})$ by $1$ .

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

Step 3

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

Tap for more steps...

Step 3.1

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

$\sec^{2} (\frac{x}{2}) - \frac{d}{d x} [x]$

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $1$ .

$\sec^{2} (\frac{x}{2}) - 1$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Reorder terms.

$- 1 + \sec^{2} (\frac{x}{2})$

Step 4.2

Reorder $- 1$ and $\sec^{2} (\frac{x}{2})$ .

$\sec^{2} (\frac{x}{2}) - 1$

Step 4.3

Apply pythagorean identity.

$\tan^{2} (\frac{x}{2})$