Calculus Examples

$y = \frac{6 x}{5 - \tan (x)}$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 x}{5 - \tan (x)}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{x}{5 - \tan (x)}]$ .

$6 \frac{d}{d x} [\frac{x}{5 - \tan (x)}]$

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 $g (x) = 5 - \tan (x)$ .

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

Step 3

Differentiate.

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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 $n = 1$ .

$6 \frac{(5 - \tan (x)) \cdot 1 - x \frac{d}{d x} [5 - \tan (x)]}{{(5 - \tan (x))}^{2}}$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$6 \frac{5 - \tan (x) - x (0 + \frac{d}{d x} [- \tan (x)])}{{(5 - \tan (x))}^{2}}$

Step 3.5

Add $0$ and $\frac{d}{d x} [- \tan (x)]$ .

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

Step 3.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- \tan (x)$ with respect to $x$ is $- \frac{d}{d x} [\tan (x)]$ .

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

Step 3.7

Multiply.

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Step 3.7.1

Multiply $- 1$ by $- 1$ .

$6 \frac{5 - \tan (x) + 1 x \frac{d}{d x} [\tan (x)]}{{(5 - \tan (x))}^{2}}$

Step 3.7.2

Multiply $x$ by $1$ .

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

Step 4

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

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

Step 5

Combine $6$ and $\frac{5 - \tan (x) + x \sec^{2} (x)}{{(5 - \tan (x))}^{2}}$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{6 \cdot 5 + 6 (- \tan (x)) + 6 (x \sec^{2} (x))}{{(5 - \tan (x))}^{2}}$

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $6$ by $5$ .

$\frac{30 + 6 (- \tan (x)) + 6 (x \sec^{2} (x))}{{(5 - \tan (x))}^{2}}$

Step 6.2.2

Multiply $- 1$ by $6$ .

$\frac{30 - 6 \tan (x) + 6 x \sec^{2} (x)}{{(5 - \tan (x))}^{2}}$

Step 6.3

Reorder terms.

$\frac{6 x \sec^{2} (x) + 30 - 6 \tan (x)}{{(5 - \tan (x))}^{2}}$

Step 6.4

Factor $6$ out of $6 x \sec^{2} (x) + 30 - 6 \tan (x)$ .

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Step 6.4.1

Factor $6$ out of $6 x \sec^{2} (x)$ .

$\frac{6 (x \sec^{2} (x)) + 30 - 6 \tan (x)}{{(5 - \tan (x))}^{2}}$

Step 6.4.2

Factor $6$ out of $30$ .

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

Step 6.4.3

Factor $6$ out of $- 6 \tan (x)$ .

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

Step 6.4.4

Factor $6$ out of $6 (x \sec^{2} (x)) + 6 (5)$ .

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

Step 6.4.5

Factor $6$ out of $6 (x \sec^{2} (x) + 5) + 6 (- \tan (x))$ .

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