Calculus Examples

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

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \tan^{2} (x)]$

Step 2

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

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Step 2.1

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

$0 - \frac{d}{d x} [\tan^{2} (x)]$

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 $f (x) = x^{2}$ and $g (x) = \tan (x)$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\tan (x)$ .

$0 - (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\tan (x)])$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$0 - (2 u \frac{d}{d x} [\tan (x)])$

Step 2.2.3

Replace all occurrences of $u$ with $\tan (x)$ .

$0 - (2 \tan (x) \frac{d}{d x} [\tan (x)])$

Step 2.3

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

$0 - (2 \tan (x) \sec^{2} (x))$

Step 2.4

Multiply $2$ by $- 1$ .

$0 - 2 \tan (x) \sec^{2} (x)$

Step 3

Simplify.

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Step 3.1

Subtract $2 \tan (x) \sec^{2} (x)$ from $0$ .

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

Step 3.2

Reorder the factors of $- 2 \tan (x) \sec^{2} (x)$ .

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