Calculus Examples

$s = 10 + 64 t - 16 t^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (10 + 64 t - 16 t^{2})$

Step 2

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

$0$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $10 + 64 t - 16 t^{2}$ with respect to $t$ is $\frac{d}{d t} [10] + \frac{d}{d t} [64 t] + \frac{d}{d t} [- 16 t^{2}]$ .

$\frac{d}{d t} [10] + \frac{d}{d t} [64 t] + \frac{d}{d t} [- 16 t^{2}]$

Step 3.1.2

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

$0 + \frac{d}{d t} [64 t] + \frac{d}{d t} [- 16 t^{2}]$

Step 3.2

Evaluate $\frac{d}{d t} [64 t]$ .

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Step 3.2.1

Since $64$ is constant with respect to $t$ , the derivative of $64 t$ with respect to $t$ is $64 \frac{d}{d t} [t]$ .

$0 + 64 \frac{d}{d t} [t] + \frac{d}{d t} [- 16 t^{2}]$

Step 3.2.2

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

$0 + 64 \cdot 1 + \frac{d}{d t} [- 16 t^{2}]$

Step 3.2.3

Multiply $64$ by $1$ .

$0 + 64 + \frac{d}{d t} [- 16 t^{2}]$

Step 3.3

Evaluate $\frac{d}{d t} [- 16 t^{2}]$ .

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Step 3.3.1

Since $- 16$ is constant with respect to $t$ , the derivative of $- 16 t^{2}$ with respect to $t$ is $- 16 \frac{d}{d t} [t^{2}]$ .

$0 + 64 - 16 \frac{d}{d t} [t^{2}]$

Step 3.3.2

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

$0 + 64 - 16 (2 t)$

Step 3.3.3

Multiply $2$ by $- 16$ .

$0 + 64 - 32 t$

Step 3.4

Simplify.

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Step 3.4.1

Add $0$ and $64$ .

$64 - 32 t$

Step 3.4.2

Reorder terms.

$- 32 t + 64$

Step 4

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

$0 = - 32 t + 64$

Step 5

Solve for $t$ .

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Step 5.1

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 32 t + 64 = 0$

Step 5.2

Subtract $64$ from both sides of the equation.

$- 32 t = - 64$

Step 5.3

Divide each term in $- 32 t = - 64$ by $- 32$ and simplify.

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Step 5.3.1

Divide each term in $- 32 t = - 64$ by $- 32$ .

$\frac{- 32 t}{- 32} = \frac{- 64}{- 32}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 32$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 32 t}{- 32} = \frac{- 64}{- 32}$

Step 5.3.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 64}{- 32}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $- 64$ by $- 32$ .

$t = 2$

Step 6

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

$\frac{d S}{d t} = 2$