Calculus Examples

$y = \frac{2 x}{7 - \tan (x)}$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 x}{7 - \tan (x)}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x}{7 - \tan (x)}]$ .

$2 \frac{d}{d x} [\frac{x}{7 - \tan (x)}]$

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 $g (x) = 7 - \tan (x)$ .

$2 \frac{(7 - \tan (x)) \frac{d}{d x} [x] - x \frac{d}{d x} [7 - \tan (x)]}{{(7 - \tan (x))}^{2}}$

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 $n = 1$ .

$2 \frac{(7 - \tan (x)) \cdot 1 - x \frac{d}{d x} [7 - \tan (x)]}{{(7 - \tan (x))}^{2}}$

Step 3.2

Multiply $7 - \tan (x)$ by $1$ .

$2 \frac{7 - \tan (x) - x \frac{d}{d x} [7 - \tan (x)]}{{(7 - \tan (x))}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $7 - \tan (x)$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- \tan (x)]$ .

$2 \frac{7 - \tan (x) - x (\frac{d}{d x} [7] + \frac{d}{d x} [- \tan (x)])}{{(7 - \tan (x))}^{2}}$

Step 3.4

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$2 \frac{7 - \tan (x) - x (0 + \frac{d}{d x} [- \tan (x)])}{{(7 - \tan (x))}^{2}}$

Step 3.5

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

$2 \frac{7 - \tan (x) - x \frac{d}{d x} [- \tan (x)]}{{(7 - \tan (x))}^{2}}$

Step 3.6

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

$2 \frac{7 - \tan (x) - x (- \frac{d}{d x} [\tan (x)])}{{(7 - \tan (x))}^{2}}$

Step 3.7

Multiply.

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Step 3.7.1

Multiply $- 1$ by $- 1$ .

$2 \frac{7 - \tan (x) + 1 x \frac{d}{d x} [\tan (x)]}{{(7 - \tan (x))}^{2}}$

Step 3.7.2

Multiply $x$ by $1$ .

$2 \frac{7 - \tan (x) + x \frac{d}{d x} [\tan (x)]}{{(7 - \tan (x))}^{2}}$

Step 4

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

$2 \frac{7 - \tan (x) + x \sec^{2} (x)}{{(7 - \tan (x))}^{2}}$

Step 5

Combine $2$ and $\frac{7 - \tan (x) + x \sec^{2} (x)}{{(7 - \tan (x))}^{2}}$ .

$\frac{2 (7 - \tan (x) + x \sec^{2} (x))}{{(7 - \tan (x))}^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{2 \cdot 7 + 2 (- \tan (x)) + 2 (x \sec^{2} (x))}{{(7 - \tan (x))}^{2}}$

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $2$ by $7$ .

$\frac{14 + 2 (- \tan (x)) + 2 (x \sec^{2} (x))}{{(7 - \tan (x))}^{2}}$

Step 6.2.2

Multiply $- 1$ by $2$ .

$\frac{14 - 2 \tan (x) + 2 x \sec^{2} (x)}{{(7 - \tan (x))}^{2}}$

Step 6.3

Reorder terms.

$\frac{2 x \sec^{2} (x) + 14 - 2 \tan (x)}{{(7 - \tan (x))}^{2}}$

Step 6.4

Factor $2$ out of $2 x \sec^{2} (x) + 14 - 2 \tan (x)$ .

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Step 6.4.1

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

$\frac{2 (x \sec^{2} (x)) + 14 - 2 \tan (x)}{{(7 - \tan (x))}^{2}}$

Step 6.4.2

Factor $2$ out of $14$ .

$\frac{2 (x \sec^{2} (x)) + 2 (7) - 2 \tan (x)}{{(7 - \tan (x))}^{2}}$

Step 6.4.3

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

$\frac{2 (x \sec^{2} (x)) + 2 (7) + 2 (- \tan (x))}{{(7 - \tan (x))}^{2}}$

Step 6.4.4

Factor $2$ out of $2 (x \sec^{2} (x)) + 2 (7)$ .

$\frac{2 (x \sec^{2} (x) + 7) + 2 (- \tan (x))}{{(7 - \tan (x))}^{2}}$

Step 6.4.5

Factor $2$ out of $2 (x \sec^{2} (x) + 7) + 2 (- \tan (x))$ .

$\frac{2 (x \sec^{2} (x) + 7 - \tan (x))}{{(7 - \tan (x))}^{2}}$