Calculus Examples

$h (t) = \frac{- 0.05 t^{2}}{2 t + 1}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move the negative in front of the fraction.

$\frac{d}{d t} [- \frac{0.05 t^{2}}{2 t + 1}]$

Step 1.2

Since $- 0.05$ is constant with respect to $t$ , the derivative of $- \frac{0.05 t^{2}}{2 t + 1}$ with respect to $t$ is $- 0.05 \frac{d}{d t} [\frac{t^{2}}{2 t + 1}]$ .

$- 0.05 \frac{d}{d t} [\frac{t^{2}}{2 t + 1}]$

Step 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 $f (t) = t^{2}$ and $g (t) = 2 t + 1$ .

$- 0.05 \frac{(2 t + 1) \frac{d}{d t} [t^{2}] - t^{2} \frac{d}{d t} [2 t + 1]}{{(2 t + 1)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$- 0.05 \frac{(2 t + 1) (2 t) - t^{2} \frac{d}{d t} [2 t + 1]}{{(2 t + 1)}^{2}}$

Step 3.2

Move $2$ to the left of $2 t + 1$ .

$- 0.05 \frac{2 \cdot (2 t + 1) t - t^{2} \frac{d}{d t} [2 t + 1]}{{(2 t + 1)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $2 t + 1$ with respect to $t$ is $\frac{d}{d t} [2 t] + \frac{d}{d t} [1]$ .

$- 0.05 \frac{2 (2 t + 1) t - t^{2} (\frac{d}{d t} [2 t] + \frac{d}{d t} [1])}{{(2 t + 1)}^{2}}$

Step 3.4

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

$- 0.05 \frac{2 (2 t + 1) t - t^{2} (2 \frac{d}{d t} [t] + \frac{d}{d t} [1])}{{(2 t + 1)}^{2}}$

Step 3.5

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

$- 0.05 \frac{2 (2 t + 1) t - t^{2} (2 \cdot 1 + \frac{d}{d t} [1])}{{(2 t + 1)}^{2}}$

Step 3.6

Multiply $2$ by $1$ .

$- 0.05 \frac{2 (2 t + 1) t - t^{2} (2 + \frac{d}{d t} [1])}{{(2 t + 1)}^{2}}$

Step 3.7

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

$- 0.05 \frac{2 (2 t + 1) t - t^{2} (2 + 0)}{{(2 t + 1)}^{2}}$

Step 3.8

Combine fractions.

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Step 3.8.1

Add $2$ and $0$ .

$- 0.05 \frac{2 (2 t + 1) t - t^{2} \cdot 2}{{(2 t + 1)}^{2}}$

Step 3.8.2

Multiply $2$ by $- 1$ .

$- 0.05 \frac{2 (2 t + 1) t - 2 t^{2}}{{(2 t + 1)}^{2}}$

Step 3.8.3

Combine $- 0.05$ and $\frac{2 (2 t + 1) t - 2 t^{2}}{{(2 t + 1)}^{2}}$ .

$\frac{- 0.05 (2 (2 t + 1) t - 2 t^{2})}{{(2 t + 1)}^{2}}$

Step 3.8.4

Move the negative in front of the fraction.

$- \frac{0.05 (2 (2 t + 1) t - 2 t^{2})}{{(2 t + 1)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$- \frac{0.05 ((2 (2 t) + 2 \cdot 1) t - 2 t^{2})}{{(2 t + 1)}^{2}}$

Step 4.2

Apply the distributive property.

$- \frac{0.05 (2 (2 t) t + 2 \cdot 1 t - 2 t^{2})}{{(2 t + 1)}^{2}}$

Step 4.3

Apply the distributive property.

$- \frac{0.05 (2 (2 t) t) + 0.05 (2 \cdot 1 t) + 0.05 (- 2 t^{2})}{{(2 t + 1)}^{2}}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

Multiply $t$ by $t$ by adding the exponents.

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Step 4.4.1.1.1

Move $t$ .

$- \frac{0.05 (2 (2 (t \cdot t))) + 0.05 (2 \cdot 1 t) + 0.05 (- 2 t^{2})}{{(2 t + 1)}^{2}}$

Step 4.4.1.1.2

Multiply $t$ by $t$ .

$- \frac{0.05 (2 (2 t^{2})) + 0.05 (2 \cdot 1 t) + 0.05 (- 2 t^{2})}{{(2 t + 1)}^{2}}$

Step 4.4.1.2

Multiply $2$ by $2$ .

$- \frac{0.05 (4 t^{2}) + 0.05 (2 \cdot 1 t) + 0.05 (- 2 t^{2})}{{(2 t + 1)}^{2}}$

Step 4.4.1.3

Multiply $4$ by $0.05$ .

$- \frac{0.2 t^{2} + 0.05 (2 \cdot 1 t) + 0.05 (- 2 t^{2})}{{(2 t + 1)}^{2}}$

Step 4.4.1.4

Multiply $2$ by $1$ .

$- \frac{0.2 t^{2} + 0.05 (2 t) + 0.05 (- 2 t^{2})}{{(2 t + 1)}^{2}}$

Step 4.4.1.5

Multiply $2$ by $0.05$ .

$- \frac{0.2 t^{2} + 0.1 t + 0.05 (- 2 t^{2})}{{(2 t + 1)}^{2}}$

Step 4.4.1.6

Multiply $- 2$ by $0.05$ .

$- \frac{0.2 t^{2} + 0.1 t - 0.1 t^{2}}{{(2 t + 1)}^{2}}$

Step 4.4.2

Subtract $0.1 t^{2}$ from $0.2 t^{2}$ .

$- \frac{0.1 t^{2} + 0.1 t}{{(2 t + 1)}^{2}}$

Step 4.5

Factor $0.1 t$ out of $0.1 t^{2} + 0.1 t$ .

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Step 4.5.1

Factor $0.1 t$ out of $0.1 t^{2}$ .

$- \frac{0.1 t (t) + 0.1 t}{{(2 t + 1)}^{2}}$

Step 4.5.2

Factor $0.1 t$ out of $0.1 t$ .

$- \frac{0.1 t (t) + 0.1 t (1)}{{(2 t + 1)}^{2}}$

Step 4.5.3

Factor $0.1 t$ out of $0.1 t (t) + 0.1 t (1)$ .

$- \frac{0.1 t (t + 1)}{{(2 t + 1)}^{2}}$