Calculus Examples

$5 x \sec (x) + x \tan (x)$

Step 1

By the Sum Rule, the derivative of $5 x \sec (x) + x \tan (x)$ with respect to $x$ is $\frac{d}{d x} [5 x \sec (x)] + \frac{d}{d x} [x \tan (x)]$ .

$\frac{d}{d x} [5 x \sec (x)] + \frac{d}{d x} [x \tan (x)]$

Step 2

Evaluate $\frac{d}{d x} [5 x \sec (x)]$ .

Tap for more steps...

Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x \sec (x)$ with respect to $x$ is $5 \frac{d}{d x} [x \sec (x)]$ .

$5 \frac{d}{d x} [x \sec (x)] + \frac{d}{d x} [x \tan (x)]$

Step 2.2

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

$5 (x \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [x]) + \frac{d}{d x} [x \tan (x)]$

Step 2.3

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

$5 (x (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [x]) + \frac{d}{d x} [x \tan (x)]$

Step 2.4

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

$5 (x (\sec (x) \tan (x)) + \sec (x) \cdot 1) + \frac{d}{d x} [x \tan (x)]$

Step 2.5

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

$5 (x (\sec (x) \tan (x)) + \sec (x)) + \frac{d}{d x} [x \tan (x)]$

Step 3

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

Tap for more steps...

Step 3.1

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 $f (x) = x$ and $g (x) = \tan (x)$ .

$5 (x (\sec (x) \tan (x)) + \sec (x)) + x \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [x]$

Step 3.2

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

$5 (x (\sec (x) \tan (x)) + \sec (x)) + x \sec^{2} (x) + \tan (x) \frac{d}{d x} [x]$

Step 3.3

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

$5 (x (\sec (x) \tan (x)) + \sec (x)) + x \sec^{2} (x) + \tan (x) \cdot 1$

Step 3.4

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

$5 (x (\sec (x) \tan (x)) + \sec (x)) + x \sec^{2} (x) + \tan (x)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

$5 (x (\sec (x) \tan (x))) + 5 \sec (x) + x \sec^{2} (x) + \tan (x)$

Step 4.2

Remove unnecessary parentheses.

$5 x (\sec (x) \tan (x)) + 5 \sec (x) + x \sec^{2} (x) + \tan (x)$

Step 4.3

Reorder terms.

$5 x \sec (x) \tan (x) + x \sec^{2} (x) + 5 \sec (x) + \tan (x)$