Calculus Examples

$y = 5 \sec (3 x) + 5 x^{3}$

Step 1

By the Sum Rule, the derivative of $5 \sec (3 x) + 5 x^{3}$ with respect to $x$ is $\frac{d}{d x} [5 \sec (3 x)] + \frac{d}{d x} [5 x^{3}]$ .

$\frac{d}{d x} [5 \sec (3 x)] + \frac{d}{d x} [5 x^{3}]$

Step 2

Evaluate $\frac{d}{d x} [5 \sec (3 x)]$ .

Tap for more steps...

Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \sec (3 x)$ with respect to $x$ is $5 \frac{d}{d x} [\sec (3 x)]$ .

$5 \frac{d}{d x} [\sec (3 x)] + \frac{d}{d x} [5 x^{3}]$

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 $f (x) = \sec (x)$ and $g (x) = 3 x$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u$ as $3 x$ .

$5 (\frac{d}{d u} [\sec (u)] \frac{d}{d x} [3 x]) + \frac{d}{d x} [5 x^{3}]$

Step 2.2.2

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

$5 (\sec (u) \tan (u) \frac{d}{d x} [3 x]) + \frac{d}{d x} [5 x^{3}]$

Step 2.2.3

Replace all occurrences of $u$ with $3 x$ .

$5 (\sec (3 x) \tan (3 x) \frac{d}{d x} [3 x]) + \frac{d}{d x} [5 x^{3}]$

Step 2.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$5 (\sec (3 x) \tan (3 x) (3 \frac{d}{d x} [x])) + \frac{d}{d x} [5 x^{3}]$

Step 2.4

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

$5 (\sec (3 x) \tan (3 x) (3 \cdot 1)) + \frac{d}{d x} [5 x^{3}]$

Step 2.5

Multiply $3$ by $1$ .

$5 (\sec (3 x) \tan (3 x) \cdot 3) + \frac{d}{d x} [5 x^{3}]$

Step 2.6

Move $3$ to the left of $\sec (3 x) \tan (3 x)$ .

$5 (3 \cdot (\sec (3 x) \tan (3 x))) + \frac{d}{d x} [5 x^{3}]$

Step 2.7

Multiply $3$ by $5$ .

$15 \sec (3 x) \tan (3 x) + \frac{d}{d x} [5 x^{3}]$

Step 3

Evaluate $\frac{d}{d x} [5 x^{3}]$ .

Tap for more steps...

Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

$15 \sec (3 x) \tan (3 x) + 5 \frac{d}{d x} [x^{3}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$15 \sec (3 x) \tan (3 x) + 5 (3 x^{2})$

Step 3.3

Multiply $3$ by $5$ .

$15 \sec (3 x) \tan (3 x) + 15 x^{2}$

Step 4

Reorder terms.

$15 x^{2} + 15 \sec (3 x) \tan (3 x)$