Calculus Examples

$f (x) = \frac{4 x^{2} \tan (x)}{\sec (x)}$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 x^{2} \tan (x)}{\sec (x)}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x^{2} \tan (x)}{\sec (x)}]$ .

$4 \frac{d}{d x} [\frac{x^{2} \tan (x)}{\sec (x)}]$

Step 2

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

$4 \frac{\sec (x) \frac{d}{d x} [x^{2} \tan (x)] - x^{2} \tan (x) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

Step 3

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

$4 \frac{\sec (x) (x^{2} \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [x^{2}]) - x^{2} \tan (x) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

Step 4

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

$4 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) \frac{d}{d x} [x^{2}]) - x^{2} \tan (x) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

Step 5

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

$4 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} \tan (x) \frac{d}{d x} [\sec (x)]}{\sec^{2} (x)}$

Step 6

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

$4 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} \tan (x) (\sec (x) \tan (x))}{\sec^{2} (x)}$

Step 7

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

$4 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} (\tan^{1} (x) \tan (x)) \sec (x)}{\sec^{2} (x)}$

Step 8

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

$4 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} (\tan^{1} (x) \tan^{1} (x)) \sec (x)}{\sec^{2} (x)}$

Step 9

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

$4 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} {\tan (x)}^{1 + 1} \sec (x)}{\sec^{2} (x)}$

Step 10

Add $1$ and $1$ .

$4 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} \tan^{2} (x) \sec (x)}{\sec^{2} (x)}$

Step 11

Factor $\sec (x)$ out of $\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) - x^{2} \tan^{2} (x) \sec (x)$ .

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Step 11.1

Factor $\sec (x)$ out of $- x^{2} \tan^{2} (x) \sec (x)$ .

$4 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) + \sec (x) (- x^{2} \tan^{2} (x))}{\sec^{2} (x)}$

Step 11.2

Factor $\sec (x)$ out of $\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x)) + \sec (x) (- x^{2} \tan^{2} (x))$ .

$4 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x))}{\sec^{2} (x)}$

Step 12

Cancel the common factors.

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Step 12.1

Factor $\sec (x)$ out of $\sec^{2} (x)$ .

$4 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x))}{\sec (x) \sec (x)}$

Step 12.2

Cancel the common factor.

$4 \frac{\sec (x) (x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x))}{\sec (x) \sec (x)}$

Step 12.3

Rewrite the expression.

$4 \frac{x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x)}{\sec (x)}$

Step 13

Combine $4$ and $\frac{x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x)}{\sec (x)}$ .

$\frac{4 (x^{2} \sec^{2} (x) + \tan (x) (2 x) - x^{2} \tan^{2} (x))}{\sec (x)}$

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

$\frac{4 (x^{2} \sec^{2} (x)) + 4 (\tan (x) (2 x)) + 4 (- x^{2} \tan^{2} (x))}{\sec (x)}$

Step 14.2

Simplify the numerator.

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Step 14.2.1

Factor $4 x$ out of $4 (x^{2} \sec^{2} (x)) + 4 (\tan (x) (2 x)) + 4 (- x^{2} \tan^{2} (x))$ .

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Step 14.2.1.1

Factor $4 x$ out of $4 (x^{2} \sec^{2} (x))$ .

$\frac{4 x (x \sec^{2} (x)) + 4 (\tan (x) (2 x)) + 4 (- x^{2} \tan^{2} (x))}{\sec (x)}$

Step 14.2.1.2

Factor $4 x$ out of $4 (\tan (x) (2 x))$ .

$\frac{4 x (x \sec^{2} (x)) + 4 x (\tan (x) \cdot (2)) + 4 (- x^{2} \tan^{2} (x))}{\sec (x)}$

Step 14.2.1.3

Factor $4 x$ out of $4 (- x^{2} \tan^{2} (x))$ .

$\frac{4 x (x \sec^{2} (x)) + 4 x (\tan (x) \cdot (2)) + 4 x (- x \tan^{2} (x))}{\sec (x)}$

Step 14.2.1.4

Factor $4 x$ out of $4 x (x \sec^{2} (x)) + 4 x (\tan (x) \cdot (2))$ .

$\frac{4 x (x \sec^{2} (x) + \tan (x) \cdot (2)) + 4 x (- x \tan^{2} (x))}{\sec (x)}$

Step 14.2.1.5

Factor $4 x$ out of $4 x (x \sec^{2} (x) + \tan (x) \cdot (2)) + 4 x (- x \tan^{2} (x))$ .

$\frac{4 x (x \sec^{2} (x) + \tan (x) \cdot (2) - x \tan^{2} (x))}{\sec (x)}$

Step 14.2.2

Move $- x \tan^{2} (x)$ .

$\frac{4 x (x \sec^{2} (x) - x \tan^{2} (x) + \tan (x) \cdot (2))}{\sec (x)}$

Step 14.2.3

Factor $x$ out of $x \sec^{2} (x)$ .

$\frac{4 x (x (\sec^{2} (x)) - x \tan^{2} (x) + \tan (x) \cdot (2))}{\sec (x)}$

Step 14.2.4

Factor $x$ out of $- x \tan^{2} (x)$ .

$\frac{4 x (x (\sec^{2} (x)) + x (- 1 \tan^{2} (x)) + \tan (x) \cdot (2))}{\sec (x)}$

Step 14.2.5

Factor $x$ out of $x (\sec^{2} (x)) + x (- 1 \tan^{2} (x))$ .

$\frac{4 x (x (\sec^{2} (x) - 1 \tan^{2} (x)) + \tan (x) \cdot (2))}{\sec (x)}$

Step 14.2.6

Apply pythagorean identity.

$\frac{4 x (x \cdot 1 + \tan (x) \cdot (2))}{\sec (x)}$

Step 14.2.7

Simplify each term.

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Step 14.2.7.1

Multiply $x$ by $1$ .

$\frac{4 x (x + \tan (x) \cdot (2))}{\sec (x)}$

Step 14.2.7.2

Move $2$ to the left of $\tan (x)$ .

$\frac{4 x (x + 2 \tan (x))}{\sec (x)}$

Step 14.2.8

Apply the distributive property.

$\frac{4 x \cdot x + 4 x (2 \tan (x))}{\sec (x)}$

Step 14.2.9

Multiply $x$ by $x$ by adding the exponents.

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Step 14.2.9.1

Move $x$ .

$\frac{4 (x \cdot x) + 4 x (2 \tan (x))}{\sec (x)}$

Step 14.2.9.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 4 x (2 \tan (x))}{\sec (x)}$

Step 14.2.10

Multiply $2$ by $4$ .

$\frac{4 x^{2} + 8 x \tan (x)}{\sec (x)}$

Step 14.3

Factor $4 x$ out of $4 x^{2} + 8 x \tan (x)$ .

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Step 14.3.1

Factor $4 x$ out of $4 x^{2}$ .

$\frac{4 x (x) + 8 x \tan (x)}{\sec (x)}$

Step 14.3.2

Factor $4 x$ out of $8 x \tan (x)$ .

$\frac{4 x (x) + 4 x (2 \tan (x))}{\sec (x)}$

Step 14.3.3

Factor $4 x$ out of $4 x (x) + 4 x (2 \tan (x))$ .

$\frac{4 x (x + 2 \tan (x))}{\sec (x)}$