Calculus Examples

$y = \frac{7 \tan (t)}{5 + 5 \sec (t)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (y) = \frac{d}{d t} (\frac{7 \tan (t)}{5 + 5 \sec (t)})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $7$ is constant with respect to $t$ , the derivative of $\frac{7 \tan (t)}{5 + 5 \sec (t)}$ with respect to $t$ is $7 \frac{d}{d t} [\frac{\tan (t)}{5 + 5 \sec (t)}]$ .

$7 \frac{d}{d t} [\frac{\tan (t)}{5 + 5 \sec (t)}]$

Step 3.2

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 $f (t) = \tan (t)$ and $g (t) = 5 + 5 \sec (t)$ .

$7 \frac{(5 + 5 \sec (t)) \frac{d}{d t} [\tan (t)] - \tan (t) \frac{d}{d t} [5 + 5 \sec (t)]}{{(5 + 5 \sec (t))}^{2}}$

Step 3.3

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

$7 \frac{(5 + 5 \sec (t)) \sec^{2} (t) - \tan (t) \frac{d}{d t} [5 + 5 \sec (t)]}{{(5 + 5 \sec (t))}^{2}}$

Step 3.4

Differentiate.

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Step 3.4.1

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

$7 \frac{(5 + 5 \sec (t)) \sec^{2} (t) - \tan (t) (\frac{d}{d t} [5] + \frac{d}{d t} [5 \sec (t)])}{{(5 + 5 \sec (t))}^{2}}$

Step 3.4.2

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

$7 \frac{(5 + 5 \sec (t)) \sec^{2} (t) - \tan (t) (0 + \frac{d}{d t} [5 \sec (t)])}{{(5 + 5 \sec (t))}^{2}}$

Step 3.4.3

Add $0$ and $\frac{d}{d t} [5 \sec (t)]$ .

$7 \frac{(5 + 5 \sec (t)) \sec^{2} (t) - \tan (t) \frac{d}{d t} [5 \sec (t)]}{{(5 + 5 \sec (t))}^{2}}$

Step 3.4.4

Since $5$ is constant with respect to $t$ , the derivative of $5 \sec (t)$ with respect to $t$ is $5 \frac{d}{d t} [\sec (t)]$ .

$7 \frac{(5 + 5 \sec (t)) \sec^{2} (t) - \tan (t) (5 \frac{d}{d t} [\sec (t)])}{{(5 + 5 \sec (t))}^{2}}$

Step 3.4.5

Multiply $5$ by $- 1$ .

$7 \frac{(5 + 5 \sec (t)) \sec^{2} (t) - 5 \tan (t) \frac{d}{d t} [\sec (t)]}{{(5 + 5 \sec (t))}^{2}}$

Step 3.5

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

$7 \frac{(5 + 5 \sec (t)) \sec^{2} (t) - 5 \tan (t) (\sec (t) \tan (t))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.6

Raise $\tan (t)$ to the power of $1$ .

$7 \frac{(5 + 5 \sec (t)) \sec^{2} (t) - 5 (\tan^{1} (t) \tan (t)) \sec (t)}{{(5 + 5 \sec (t))}^{2}}$

Step 3.7

Raise $\tan (t)$ to the power of $1$ .

$7 \frac{(5 + 5 \sec (t)) \sec^{2} (t) - 5 (\tan^{1} (t) \tan^{1} (t)) \sec (t)}{{(5 + 5 \sec (t))}^{2}}$

Step 3.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$7 \frac{(5 + 5 \sec (t)) \sec^{2} (t) - 5 {\tan (t)}^{1 + 1} \sec (t)}{{(5 + 5 \sec (t))}^{2}}$

Step 3.9

Add $1$ and $1$ .

$7 \frac{(5 + 5 \sec (t)) \sec^{2} (t) - 5 \tan^{2} (t) \sec (t)}{{(5 + 5 \sec (t))}^{2}}$

Step 3.10

Combine $7$ and $\frac{(5 + 5 \sec (t)) \sec^{2} (t) - 5 \tan^{2} (t) \sec (t)}{{(5 + 5 \sec (t))}^{2}}$ .

$\frac{7 ((5 + 5 \sec (t)) \sec^{2} (t) - 5 \tan^{2} (t) \sec (t))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

$\frac{7 (5 \sec^{2} (t) + 5 \sec (t) \sec^{2} (t) - 5 \tan^{2} (t) \sec (t))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.2

Apply the distributive property.

$\frac{7 (5 \sec^{2} (t)) + 7 (5 \sec (t) \sec^{2} (t)) + 7 (- 5 \tan^{2} (t) \sec (t))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3

Simplify the numerator.

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Step 3.11.3.1

Factor $7 \sec (t)$ out of $7 (5 \sec^{2} (t)) + 7 (5 \sec (t) \sec^{2} (t)) + 7 (- 5 \tan^{2} (t) \sec (t))$ .

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Step 3.11.3.1.1

Factor $7 \sec (t)$ out of $7 (5 \sec^{2} (t))$ .

$\frac{7 \sec (t) (5 \sec (t)) + 7 (5 \sec (t) \sec^{2} (t)) + 7 (- 5 \tan^{2} (t) \sec (t))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.1.2

Factor $7 \sec (t)$ out of $7 (5 \sec (t) \sec^{2} (t))$ .

$\frac{7 \sec (t) (5 \sec (t)) + 7 \sec (t) (5 \sec^{2} (t)) + 7 (- 5 \tan^{2} (t) \sec (t))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.1.3

Factor $7 \sec (t)$ out of $7 (- 5 \tan^{2} (t) \sec (t))$ .

$\frac{7 \sec (t) (5 \sec (t)) + 7 \sec (t) (5 \sec^{2} (t)) + 7 \sec (t) (- 5 \tan^{2} (t))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.1.4

Factor $7 \sec (t)$ out of $7 \sec (t) (5 \sec (t)) + 7 \sec (t) (5 \sec^{2} (t))$ .

$\frac{7 \sec (t) (5 \sec (t) + 5 \sec^{2} (t)) + 7 \sec (t) (- 5 \tan^{2} (t))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.1.5

Factor $7 \sec (t)$ out of $7 \sec (t) (5 \sec (t) + 5 \sec^{2} (t)) + 7 \sec (t) (- 5 \tan^{2} (t))$ .

$\frac{7 \sec (t) (5 \sec (t) + 5 \sec^{2} (t) - 5 \tan^{2} (t))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.2

Factor $5$ out of $5 \sec^{2} (t)$ .

$\frac{7 \sec (t) (5 \sec (t) + 5 (\sec^{2} (t)) - 5 \tan^{2} (t))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.3

Factor $5$ out of $- 5 \tan^{2} (t)$ .

$\frac{7 \sec (t) (5 \sec (t) + 5 (\sec^{2} (t)) + 5 (- \tan^{2} (t)))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.4

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

$\frac{7 \sec (t) (5 \sec (t) + 5 (\sec^{2} (t) - \tan^{2} (t)))}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.5

Apply pythagorean identity.

$\frac{7 \sec (t) (5 \sec (t) + 5 \cdot 1)}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.6

Multiply $5$ by $1$ .

$\frac{7 \sec (t) (5 \sec (t) + 5)}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.7

Apply the distributive property.

$\frac{7 \sec (t) (5 \sec (t)) + 7 \sec (t) \cdot 5}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.8

Multiply $7 \sec (t) (5 \sec (t))$ .

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Step 3.11.3.8.1

Multiply $5$ by $7$ .

$\frac{35 \sec (t) \sec (t) + 7 \sec (t) \cdot 5}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.8.2

Raise $\sec (t)$ to the power of $1$ .

$\frac{35 (\sec^{1} (t) \sec (t)) + 7 \sec (t) \cdot 5}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.8.3

Raise $\sec (t)$ to the power of $1$ .

$\frac{35 (\sec^{1} (t) \sec^{1} (t)) + 7 \sec (t) \cdot 5}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{35 {\sec (t)}^{1 + 1} + 7 \sec (t) \cdot 5}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.8.5

Add $1$ and $1$ .

$\frac{35 \sec^{2} (t) + 7 \sec (t) \cdot 5}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.3.9

Multiply $5$ by $7$ .

$\frac{35 \sec^{2} (t) + 35 \sec (t)}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.4

Factor $35 \sec (t)$ out of $35 \sec^{2} (t) + 35 \sec (t)$ .

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Step 3.11.4.1

Factor $35 \sec (t)$ out of $35 \sec^{2} (t)$ .

$\frac{35 \sec (t) (\sec (t)) + 35 \sec (t)}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.4.2

Factor $35 \sec (t)$ out of $35 \sec (t)$ .

$\frac{35 \sec (t) (\sec (t)) + 35 \sec (t) (1)}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.4.3

Factor $35 \sec (t)$ out of $35 \sec (t) (\sec (t)) + 35 \sec (t) (1)$ .

$\frac{35 \sec (t) (\sec (t) + 1)}{{(5 + 5 \sec (t))}^{2}}$

Step 3.11.5

Simplify the denominator.

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Step 3.11.5.1

Factor $5$ out of $5 + 5 \sec (t)$ .

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Step 3.11.5.1.1

Factor $5$ out of $5$ .

$\frac{35 \sec (t) (\sec (t) + 1)}{{(5 \cdot 1 + 5 \sec (t))}^{2}}$

Step 3.11.5.1.2

Factor $5$ out of $5 \cdot 1 + 5 \sec (t)$ .

$\frac{35 \sec (t) (\sec (t) + 1)}{{(5 (1 + \sec (t)))}^{2}}$

Step 3.11.5.2

Apply the product rule to $5 (1 + \sec (t))$ .

$\frac{35 \sec (t) (\sec (t) + 1)}{5^{2} {(1 + \sec (t))}^{2}}$

Step 3.11.5.3

Raise $5$ to the power of $2$ .

$\frac{35 \sec (t) (\sec (t) + 1)}{25 {(1 + \sec (t))}^{2}}$

Step 3.11.6

Cancel the common factor of $35$ and $25$ .

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Step 3.11.6.1

Factor $5$ out of $35 \sec (t) (\sec (t) + 1)$ .

$\frac{5 (7 \sec (t) (\sec (t) + 1))}{25 {(1 + \sec (t))}^{2}}$

Step 3.11.6.2

Cancel the common factors.

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Step 3.11.6.2.1

Factor $5$ out of $25 {(1 + \sec (t))}^{2}$ .

$\frac{5 (7 \sec (t) (\sec (t) + 1))}{5 (5 {(1 + \sec (t))}^{2})}$

Step 3.11.6.2.2

Cancel the common factor.

$\frac{5 (7 \sec (t) (\sec (t) + 1))}{5 (5 {(1 + \sec (t))}^{2})}$

Step 3.11.6.2.3

Rewrite the expression.

$\frac{7 \sec (t) (\sec (t) + 1)}{5 {(1 + \sec (t))}^{2}}$

Step 3.11.7

Cancel the common factor of $\sec (t) + 1$ and ${(1 + \sec (t))}^{2}$ .

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Step 3.11.7.1

Reorder terms.

$\frac{7 \sec (t) (1 + \sec (t))}{5 {(1 + \sec (t))}^{2}}$

Step 3.11.7.2

Factor $1 + \sec (t)$ out of $7 \sec (t) (1 + \sec (t))$ .

$\frac{(1 + \sec (t)) (7 \sec (t))}{5 {(1 + \sec (t))}^{2}}$

Step 3.11.7.3

Cancel the common factors.

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Step 3.11.7.3.1

Factor $1 + \sec (t)$ out of $5 {(1 + \sec (t))}^{2}$ .

$\frac{(1 + \sec (t)) (7 \sec (t))}{(1 + \sec (t)) (5 (1 + \sec (t)))}$

Step 3.11.7.3.2

Cancel the common factor.

$\frac{(1 + \sec (t)) (7 \sec (t))}{(1 + \sec (t)) (5 (1 + \sec (t)))}$

Step 3.11.7.3.3

Rewrite the expression.

$\frac{7 \sec (t)}{5 (1 + \sec (t))}$

Step 4

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

$y' = \frac{7 \sec (t)}{5 (1 + \sec (t))}$

Step 5

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

$\frac{d y}{d t} = \frac{7 \sec (t)}{5 (1 + \sec (t))}$