Calculus Examples

$f (x) = (9 x^{5} - 6 x^{- 2}) \sec (x)$

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 $f (x) = 9 x^{5} - 6 x^{- 2}$ and $g (x) = \sec (x)$ .

$(9 x^{5} - 6 x^{- 2}) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [9 x^{5} - 6 x^{- 2}]$

Step 2

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

$(9 x^{5} - 6 x^{- 2}) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [9 x^{5} - 6 x^{- 2}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $9 x^{5} - 6 x^{- 2}$ with respect to $x$ is $\frac{d}{d x} [9 x^{5}] + \frac{d}{d x} [- 6 x^{- 2}]$ .

$(9 x^{5} - 6 x^{- 2}) \sec (x) \tan (x) + \sec (x) (\frac{d}{d x} [9 x^{5}] + \frac{d}{d x} [- 6 x^{- 2}])$

Step 3.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{5}$ with respect to $x$ is $9 \frac{d}{d x} [x^{5}]$ .

$(9 x^{5} - 6 x^{- 2}) \sec (x) \tan (x) + \sec (x) (9 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 6 x^{- 2}])$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$(9 x^{5} - 6 x^{- 2}) \sec (x) \tan (x) + \sec (x) (9 (5 x^{4}) + \frac{d}{d x} [- 6 x^{- 2}])$

Step 3.4

Multiply $5$ by $9$ .

$(9 x^{5} - 6 x^{- 2}) \sec (x) \tan (x) + \sec (x) (45 x^{4} + \frac{d}{d x} [- 6 x^{- 2}])$

Step 3.5

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{- 2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{- 2}]$ .

$(9 x^{5} - 6 x^{- 2}) \sec (x) \tan (x) + \sec (x) (45 x^{4} - 6 \frac{d}{d x} [x^{- 2}])$

Step 3.6

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

$(9 x^{5} - 6 x^{- 2}) \sec (x) \tan (x) + \sec (x) (45 x^{4} - 6 (- 2 x^{- 3}))$

Step 3.7

Multiply $- 2$ by $- 6$ .

$(9 x^{5} - 6 x^{- 2}) \sec (x) \tan (x) + \sec (x) (45 x^{4} + 12 x^{- 3})$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(9 x^{5} - 6 \frac{1}{x^{2}}) \sec (x) \tan (x) + \sec (x) (45 x^{4} + 12 x^{- 3})$

Step 5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(9 x^{5} - 6 \frac{1}{x^{2}}) \sec (x) \tan (x) + \sec (x) (45 x^{4} + 12 \frac{1}{x^{3}})$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$(9 x^{5} \sec (x) - 6 \frac{1}{x^{2}} \sec (x)) \tan (x) + \sec (x) (45 x^{4} + 12 \frac{1}{x^{3}})$

Step 6.2

Apply the distributive property.

$9 x^{5} \sec (x) \tan (x) - 6 \frac{1}{x^{2}} \sec (x) \tan (x) + \sec (x) (45 x^{4} + 12 \frac{1}{x^{3}})$

Step 6.3

Apply the distributive property.

$9 x^{5} \sec (x) \tan (x) - 6 \frac{1}{x^{2}} \sec (x) \tan (x) + \sec (x) (45 x^{4}) + \sec (x) (12 \frac{1}{x^{3}})$

Step 6.4

Combine terms.

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Step 6.4.1

Combine $- 6$ and $\frac{1}{x^{2}}$ .

$9 x^{5} \sec (x) \tan (x) + \frac{- 6}{x^{2}} \sec (x) \tan (x) + \sec (x) (45 x^{4}) + \sec (x) (12 \frac{1}{x^{3}})$

Step 6.4.2

Move the negative in front of the fraction.

$9 x^{5} \sec (x) \tan (x) - \frac{6}{x^{2}} \sec (x) \tan (x) + \sec (x) (45 x^{4}) + \sec (x) (12 \frac{1}{x^{3}})$

Step 6.4.3

Combine $\sec (x)$ and $\frac{6}{x^{2}}$ .

$9 x^{5} \sec (x) \tan (x) - \frac{\sec (x) \cdot 6}{x^{2}} \tan (x) + \sec (x) (45 x^{4}) + \sec (x) (12 \frac{1}{x^{3}})$

Step 6.4.4

Move $6$ to the left of $\sec (x)$ .

$9 x^{5} \sec (x) \tan (x) - \frac{6 \cdot \sec (x)}{x^{2}} \tan (x) + \sec (x) (45 x^{4}) + \sec (x) (12 \frac{1}{x^{3}})$

Step 6.4.5

Combine $\tan (x)$ and $\frac{6 \sec (x)}{x^{2}}$ .

$9 x^{5} \sec (x) \tan (x) - \frac{\tan (x) (6 \sec (x))}{x^{2}} + \sec (x) (45 x^{4}) + \sec (x) (12 \frac{1}{x^{3}})$

Step 6.4.6

Move $6$ to the left of $\tan (x)$ .

$9 x^{5} \sec (x) \tan (x) - \frac{6 \cdot \tan (x) \sec (x)}{x^{2}} + \sec (x) (45 x^{4}) + \sec (x) (12 \frac{1}{x^{3}})$

Step 6.4.7

Combine $12$ and $\frac{1}{x^{3}}$ .

$9 x^{5} \sec (x) \tan (x) - \frac{6 \tan (x) \sec (x)}{x^{2}} + \sec (x) (45 x^{4}) + \sec (x) \frac{12}{x^{3}}$

Step 6.4.8

Combine $\sec (x)$ and $\frac{12}{x^{3}}$ .

$9 x^{5} \sec (x) \tan (x) - \frac{6 \tan (x) \sec (x)}{x^{2}} + \sec (x) (45 x^{4}) + \frac{\sec (x) \cdot 12}{x^{3}}$

Step 6.4.9

Move $12$ to the left of $\sec (x)$ .

$9 x^{5} \sec (x) \tan (x) - \frac{6 \tan (x) \sec (x)}{x^{2}} + \sec (x) (45 x^{4}) + \frac{12 \sec (x)}{x^{3}}$

Step 6.5

Reorder terms.

$9 x^{5} \sec (x) \tan (x) + 45 x^{4} \sec (x) - \frac{6 \tan (x) \sec (x)}{x^{2}} + \frac{12 \sec (x)}{x^{3}}$