Calculus Examples

$f (x) = \tan (x) - x$

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $1$ .

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

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

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

Step 4.3

Apply pythagorean identity.

$\tan^{2} (x)$