Calculus Examples

$\frac{5 x}{\sec (5 x)}$

Step 1

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

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

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 $f (x) = x$ and $g (x) = \sec (5 x)$ .

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

$5 \frac{\sec (5 x) \cdot 1 - x \frac{d}{d x} [\sec (5 x)]}{\sec^{2} (5 x)}$

Step 3.2

Multiply $\sec (5 x)$ by $1$ .

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

Step 4

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 $g (x) = 5 x$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $5 x$ .

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

Step 4.2

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

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

Step 4.3

Replace all occurrences of $u$ with $5 x$ .

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

Step 5

Factor $\sec (5 x)$ out of $\sec (5 x) - x (\sec (5 x) \tan (5 x) \frac{d}{d x} [5 x])$ .

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Step 5.1

Multiply by $1$ .

$5 \frac{\sec (5 x) \cdot 1 - x (\sec (5 x) \tan (5 x) \frac{d}{d x} [5 x])}{\sec^{2} (5 x)}$

Step 5.2

Factor $\sec (5 x)$ out of $- x (\sec (5 x) \tan (5 x) \frac{d}{d x} [5 x])$ .

$5 \frac{\sec (5 x) \cdot 1 + \sec (5 x) (- x (\tan (5 x) \frac{d}{d x} [5 x]))}{\sec^{2} (5 x)}$

Step 5.3

Factor $\sec (5 x)$ out of $\sec (5 x) \cdot 1 + \sec (5 x) (- x (\tan (5 x) \frac{d}{d x} [5 x]))$ .

$5 \frac{\sec (5 x) (1 - x (\tan (5 x) \frac{d}{d x} [5 x]))}{\sec^{2} (5 x)}$

Step 6

Cancel the common factors.

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Step 6.1

Factor $\sec (5 x)$ out of $\sec^{2} (5 x)$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

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

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

Step 8

Multiply $5$ by $- 1$ .

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

Step 9

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

$5 \frac{1 - 5 x (\tan (5 x) \cdot 1)}{\sec (5 x)}$

Step 10

Combine fractions.

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Step 10.1

Multiply $\tan (5 x)$ by $1$ .

$5 \frac{1 - 5 x \tan (5 x)}{\sec (5 x)}$

Step 10.2

Combine $5$ and $\frac{1 - 5 x \tan (5 x)}{\sec (5 x)}$ .

$\frac{5 (1 - 5 x \tan (5 x))}{\sec (5 x)}$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$\frac{5 \cdot 1 + 5 (- 5 x \tan (5 x))}{\sec (5 x)}$

Step 11.2

Simplify each term.

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Step 11.2.1

Multiply $5$ by $1$ .

$\frac{5 + 5 (- 5 x \tan (5 x))}{\sec (5 x)}$

Step 11.2.2

Multiply $- 5$ by $5$ .

$\frac{5 - 25 x \tan (5 x)}{\sec (5 x)}$

Step 11.3

Reorder terms.

$\frac{- 25 x \tan (5 x) + 5}{\sec (5 x)}$