Calculus Examples

$s = \frac{1 + \sec (t)}{1 - \sec (t)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (\frac{1 + \sec (t)}{1 - \sec (t)})$

Step 2

The derivative of $s$ with respect to $t$ is $s'$ .

$s'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

Multiply.

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Step 3.4.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.4.5.2

Multiply $1 + \sec (t)$ by $1$ .

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

Step 3.5

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

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

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

Apply the distributive property.

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

Step 3.6.3

Apply the distributive property.

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

Step 3.6.4

Apply the distributive property.

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

Step 3.6.5

Simplify the numerator.

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Step 3.6.5.1

Combine the opposite terms in $1 \sec (t) \tan (t) - \sec (t) \sec (t) \tan (t) + 1 \sec (t) \tan (t) + \sec (t) \sec (t) \tan (t)$ .

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Step 3.6.5.1.1

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

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

Step 3.6.5.1.2

Add $1 \sec (t) \tan (t) + 1 \sec (t) \tan (t)$ and $0$ .

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

Step 3.6.5.2

Simplify each term.

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Step 3.6.5.2.1

Multiply $\sec (t)$ by $1$ .

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

Step 3.6.5.2.2

Multiply $\sec (t)$ by $1$ .

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

Step 3.6.5.3

Add $\sec (t) \tan (t)$ and $\sec (t) \tan (t)$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

$s' = \frac{2 \sec (t) \tan (t)}{{(1 - \sec (t))}^{2}}$

Step 5

Replace $s'$ with $\frac{d s}{d t}$ .

$\frac{d s}{d t} = \frac{2 \sec (t) \tan (t)}{{(1 - \sec (t))}^{2}}$