Calculus Examples

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

Step 1

Differentiate.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

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 $g (x) = \sec (x)$ .

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$0 - (2 \sec (x) (\sec (x) \tan (x)))$

Step 2.4

Raise $\sec (x)$ to the power of $1$ .

$0 - (2 (\sec^{1} (x) \sec (x)) \tan (x))$

Step 2.5

Raise $\sec (x)$ to the power of $1$ .

$0 - (2 (\sec^{1} (x) \sec^{1} (x)) \tan (x))$

Step 2.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$0 - (2 {\sec (x)}^{1 + 1} \tan (x))$

Step 2.7

Add $1$ and $1$ .

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

Step 2.8

Multiply $2$ by $- 1$ .

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

Step 3

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

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