Calculus Examples

$10 - \frac{1}{5} {(t - 6)}^{2}$

Step 1

Rewrite ${(t - 6)}^{2}$ as $(t - 6) (t - 6)$ .

$\frac{d}{d t} [10 - \frac{1}{5} ((t - 6) (t - 6))]$

Step 2

Expand $(t - 6) (t - 6)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d t} [10 - \frac{1}{5} (t (t - 6) - 6 (t - 6))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d t} [10 - \frac{1}{5} (t \cdot t + t \cdot - 6 - 6 (t - 6))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d t} [10 - \frac{1}{5} (t \cdot t + t \cdot - 6 - 6 t - 6 \cdot - 6)]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $t$ by $t$ .

$\frac{d}{d t} [10 - \frac{1}{5} (t^{2} + t \cdot - 6 - 6 t - 6 \cdot - 6)]$

Step 3.1.2

Move $- 6$ to the left of $t$ .

$\frac{d}{d t} [10 - \frac{1}{5} (t^{2} - 6 \cdot t - 6 t - 6 \cdot - 6)]$

Step 3.1.3

Multiply $- 6$ by $- 6$ .

$\frac{d}{d t} [10 - \frac{1}{5} (t^{2} - 6 t - 6 t + 36)]$

Step 3.2

Subtract $6 t$ from $- 6 t$ .

$\frac{d}{d t} [10 - \frac{1}{5} (t^{2} - 12 t + 36)]$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $10 - \frac{1}{5} (t^{2} - 12 t + 36)$ with respect to $t$ is $\frac{d}{d t} [10] + \frac{d}{d t} [- \frac{1}{5} (t^{2} - 12 t + 36)]$ .

$\frac{d}{d t} [10] + \frac{d}{d t} [- \frac{1}{5} (t^{2} - 12 t + 36)]$

Step 4.2

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

$0 + \frac{d}{d t} [- \frac{1}{5} (t^{2} - 12 t + 36)]$

Step 5

Evaluate $\frac{d}{d t} [- \frac{1}{5} (t^{2} - 12 t + 36)]$ .

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Step 5.1

Since $- \frac{1}{5}$ is constant with respect to $t$ , the derivative of $- \frac{1}{5} (t^{2} - 12 t + 36)$ with respect to $t$ is $- \frac{1}{5} \frac{d}{d t} [t^{2} - 12 t + 36]$ .

$0 - \frac{1}{5} \frac{d}{d t} [t^{2} - 12 t + 36]$

Step 5.2

By the Sum Rule, the derivative of $t^{2} - 12 t + 36$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 12 t] + \frac{d}{d t} [36]$ .

$0 - \frac{1}{5} (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 12 t] + \frac{d}{d t} [36])$

Step 5.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$0 - \frac{1}{5} (2 t + \frac{d}{d t} [- 12 t] + \frac{d}{d t} [36])$

Step 5.4

Since $- 12$ is constant with respect to $t$ , the derivative of $- 12 t$ with respect to $t$ is $- 12 \frac{d}{d t} [t]$ .

$0 - \frac{1}{5} (2 t - 12 \frac{d}{d t} [t] + \frac{d}{d t} [36])$

Step 5.5

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$0 - \frac{1}{5} (2 t - 12 \cdot 1 + \frac{d}{d t} [36])$

Step 5.6

Since $36$ is constant with respect to $t$ , the derivative of $36$ with respect to $t$ is $0$ .

$0 - \frac{1}{5} (2 t - 12 \cdot 1 + 0)$

Step 5.7

Multiply $- 12$ by $1$ .

$0 - \frac{1}{5} (2 t - 12 + 0)$

Step 5.8

Add $2 t - 12$ and $0$ .

$0 - \frac{1}{5} (2 t - 12)$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$0 - \frac{1}{5} (2 t) - \frac{1}{5} \cdot - 12$

Step 6.2

Combine terms.

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Step 6.2.1

Multiply $2$ by $- 1$ .

$0 - 2 (\frac{1}{5}) t - \frac{1}{5} \cdot - 12$

Step 6.2.2

Combine $- 2$ and $\frac{1}{5}$ .

$0 + \frac{- 2}{5} t - \frac{1}{5} \cdot - 12$

Step 6.2.3

Combine $\frac{- 2}{5}$ and $t$ .

$0 + \frac{- 2 t}{5} - \frac{1}{5} \cdot - 12$

Step 6.2.4

Move the negative in front of the fraction.

$0 - \frac{2 t}{5} - \frac{1}{5} \cdot - 12$

Step 6.2.5

Multiply $- 12$ by $- 1$ .

$0 - \frac{2 t}{5} + 12 (\frac{1}{5})$

Step 6.2.6

Combine $12$ and $\frac{1}{5}$ .

$0 - \frac{2 t}{5} + \frac{12}{5}$

Step 6.2.7

Subtract $- (- \frac{2 t}{5} + \frac{12}{5})$ from $0$ .

$- \frac{2 t}{5} + \frac{12}{5}$