Calculus Examples

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

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} + 1$ and $g (x) = \sec (x)$ .

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

Step 2

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

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $2 x$ and $0$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Multiply $\sec (x)$ by $1$ .

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

Step 4.4

Reorder terms.

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