Calculus Examples

$3 x^{2} \sec (x)$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2} \sec (x)$ with respect to $x$ is $3 \frac{d}{d x} [x^{2} \sec (x)]$ .

$3 \frac{d}{d x} [x^{2} \sec (x)]$

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

$3 (x^{2} \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [x^{2}])$

Step 3

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

$3 (x^{2} (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [x^{2}])$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $3$ .

$3 x^{2} (\sec (x) \tan (x)) + 6 \sec (x) x$

Step 5.3

Reorder terms.

$3 x^{2} \sec (x) \tan (x) + 6 x \sec (x)$