Calculus Examples

$y = x - \tan (x)$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Factor $- 1$ out of $- \sec^{2} (x)$ .

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

Step 3.3

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.4

Factor $- 1$ out of $- \sec^{2} (x) - 1 \cdot - 1$ .

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

Step 3.5

Apply pythagorean identity.

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