Calculus Examples

$\frac{10 + \sec (q)}{10 - \sec (q)}$

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 $f (q) = 10 + \sec (q)$ and $g (q) = 10 - \sec (q)$ .

$\frac{(10 - \sec (q)) \frac{d}{d q} [10 + \sec (q)] - (10 + \sec (q)) \frac{d}{d q} [10 - \sec (q)]}{{(10 - \sec (q))}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $10 + \sec (q)$ with respect to $q$ is $\frac{d}{d q} [10] + \frac{d}{d q} [\sec (q)]$ .

$\frac{(10 - \sec (q)) (\frac{d}{d q} [10] + \frac{d}{d q} [\sec (q)]) - (10 + \sec (q)) \frac{d}{d q} [10 - \sec (q)]}{{(10 - \sec (q))}^{2}}$

Step 2.2

Since $10$ is constant with respect to $q$ , the derivative of $10$ with respect to $q$ is $0$ .

$\frac{(10 - \sec (q)) (0 + \frac{d}{d q} [\sec (q)]) - (10 + \sec (q)) \frac{d}{d q} [10 - \sec (q)]}{{(10 - \sec (q))}^{2}}$

Step 2.3

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

$\frac{(10 - \sec (q)) \frac{d}{d q} [\sec (q)] - (10 + \sec (q)) \frac{d}{d q} [10 - \sec (q)]}{{(10 - \sec (q))}^{2}}$

Step 3

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

$\frac{(10 - \sec (q)) (\sec (q) \tan (q)) - (10 + \sec (q)) \frac{d}{d q} [10 - \sec (q)]}{{(10 - \sec (q))}^{2}}$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $10 - \sec (q)$ with respect to $q$ is $\frac{d}{d q} [10] + \frac{d}{d q} [- \sec (q)]$ .

$\frac{(10 - \sec (q)) \sec (q) \tan (q) - (10 + \sec (q)) (\frac{d}{d q} [10] + \frac{d}{d q} [- \sec (q)])}{{(10 - \sec (q))}^{2}}$

Step 4.2

Since $10$ is constant with respect to $q$ , the derivative of $10$ with respect to $q$ is $0$ .

$\frac{(10 - \sec (q)) \sec (q) \tan (q) - (10 + \sec (q)) (0 + \frac{d}{d q} [- \sec (q)])}{{(10 - \sec (q))}^{2}}$

Step 4.3

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

$\frac{(10 - \sec (q)) \sec (q) \tan (q) - (10 + \sec (q)) \frac{d}{d q} [- \sec (q)]}{{(10 - \sec (q))}^{2}}$

Step 4.4

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

$\frac{(10 - \sec (q)) \sec (q) \tan (q) - (10 + \sec (q)) (- \frac{d}{d q} [\sec (q)])}{{(10 - \sec (q))}^{2}}$

Step 4.5

Multiply.

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Step 4.5.1

Multiply $- 1$ by $- 1$ .

$\frac{(10 - \sec (q)) \sec (q) \tan (q) + 1 (10 + \sec (q)) \frac{d}{d q} [\sec (q)]}{{(10 - \sec (q))}^{2}}$

Step 4.5.2

Multiply $10 + \sec (q)$ by $1$ .

$\frac{(10 - \sec (q)) \sec (q) \tan (q) + (10 + \sec (q)) \frac{d}{d q} [\sec (q)]}{{(10 - \sec (q))}^{2}}$

Step 5

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

$\frac{(10 - \sec (q)) \sec (q) \tan (q) + (10 + \sec (q)) (\sec (q) \tan (q))}{{(10 - \sec (q))}^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{(10 \sec (q) - \sec (q) \sec (q)) \tan (q) + (10 + \sec (q)) \sec (q) \tan (q)}{{(10 - \sec (q))}^{2}}$

Step 6.2

Apply the distributive property.

$\frac{10 \sec (q) \tan (q) - \sec (q) \sec (q) \tan (q) + (10 + \sec (q)) \sec (q) \tan (q)}{{(10 - \sec (q))}^{2}}$

Step 6.3

Apply the distributive property.

$\frac{10 \sec (q) \tan (q) - \sec (q) \sec (q) \tan (q) + (10 \sec (q) + \sec (q) \sec (q)) \tan (q)}{{(10 - \sec (q))}^{2}}$

Step 6.4

Apply the distributive property.

$\frac{10 \sec (q) \tan (q) - \sec (q) \sec (q) \tan (q) + 10 \sec (q) \tan (q) + \sec (q) \sec (q) \tan (q)}{{(10 - \sec (q))}^{2}}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Combine the opposite terms in $10 \sec (q) \tan (q) - \sec (q) \sec (q) \tan (q) + 10 \sec (q) \tan (q) + \sec (q) \sec (q) \tan (q)$ .

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Step 6.5.1.1

Add $- \sec (q) \sec (q) \tan (q)$ and $\sec (q) \sec (q) \tan (q)$ .

$\frac{10 \sec (q) \tan (q) + 10 \sec (q) \tan (q) + 0}{{(10 - \sec (q))}^{2}}$

Step 6.5.1.2

Add $10 \sec (q) \tan (q) + 10 \sec (q) \tan (q)$ and $0$ .

$\frac{10 \sec (q) \tan (q) + 10 \sec (q) \tan (q)}{{(10 - \sec (q))}^{2}}$

Step 6.5.2

Add $10 \sec (q) \tan (q)$ and $10 \sec (q) \tan (q)$ .

$\frac{20 \sec (q) \tan (q)}{{(10 - \sec (q))}^{2}}$