Calculus Examples

$\frac{\tan (q)}{1 + \tan (q)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d q} [\frac{f (q)}{g (q)}]$ is $\frac{g (q) \frac{d}{d q} [f (q)] - f (q) \frac{d}{d q} [g (q)]}{{g (q)}^{2}}$ where $f (q) = \tan (q)$ and $g (q) = 1 + \tan (q)$ .

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 + \tan (q)$ with respect to $q$ is $\frac{d}{d q} [1] + \frac{d}{d q} [\tan (q)]$ .

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

Step 3.2

Since $1$ is constant with respect to $q$ , the derivative of $1$ with respect to $q$ is $0$ .

$\frac{(1 + \tan (q)) \sec^{2} (q) - \tan (q) (0 + \frac{d}{d q} [\tan (q)])}{{(1 + \tan (q))}^{2}}$

Step 3.3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Simplify the numerator.

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Step 5.2.1

Combine the opposite terms in $1 \sec^{2} (q) + \tan (q) \sec^{2} (q) - \tan (q) \sec^{2} (q)$ .

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Step 5.2.1.1

Subtract $\tan (q) \sec^{2} (q)$ from $\tan (q) \sec^{2} (q)$ .

$\frac{1 \sec^{2} (q) + 0}{{(1 + \tan (q))}^{2}}$

Step 5.2.1.2

Add $1 \sec^{2} (q)$ and $0$ .

$\frac{1 \sec^{2} (q)}{{(1 + \tan (q))}^{2}}$

Step 5.2.2

Multiply $\sec^{2} (q)$ by $1$ .

$\frac{\sec^{2} (q)}{{(1 + \tan (q))}^{2}}$