Calculus Examples

$- \frac{3000 t}{(1 + t^{2})}$

Step 1

Since $- 3000$ is constant with respect to $t$ , the derivative of $- \frac{3000 t}{1 + t^{2}}$ with respect to $t$ is $- 3000 \frac{d}{d t} [\frac{t}{1 + t^{2}}]$ .

$- 3000 \frac{d}{d t} [\frac{t}{1 + t^{2}}]$

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

$- 3000 \frac{(1 + t^{2}) \frac{d}{d t} [t] - t \frac{d}{d t} [1 + t^{2}]}{{(1 + t^{2})}^{2}}$

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 $n = 1$ .

$- 3000 \frac{(1 + t^{2}) \cdot 1 - t \frac{d}{d t} [1 + t^{2}]}{{(1 + t^{2})}^{2}}$

Step 3.2

Multiply $1 + t^{2}$ by $1$ .

$- 3000 \frac{1 + t^{2} - t \frac{d}{d t} [1 + t^{2}]}{{(1 + t^{2})}^{2}}$

Step 3.3

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

$- 3000 \frac{1 + t^{2} - t (\frac{d}{d t} [1] + \frac{d}{d t} [t^{2}])}{{(1 + t^{2})}^{2}}$

Step 3.4

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

$- 3000 \frac{1 + t^{2} - t (0 + \frac{d}{d t} [t^{2}])}{{(1 + t^{2})}^{2}}$

Step 3.5

Add $0$ and $\frac{d}{d t} [t^{2}]$ .

$- 3000 \frac{1 + t^{2} - t \frac{d}{d t} [t^{2}]}{{(1 + t^{2})}^{2}}$

Step 3.6

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

$- 3000 \frac{1 + t^{2} - t (2 t)}{{(1 + t^{2})}^{2}}$

Step 3.7

Multiply $2$ by $- 1$ .

$- 3000 \frac{1 + t^{2} - 2 t \cdot t}{{(1 + t^{2})}^{2}}$

Step 4

Raise $t$ to the power of $1$ .

$- 3000 \frac{1 + t^{2} - 2 (t^{1} t)}{{(1 + t^{2})}^{2}}$

Step 5

Raise $t$ to the power of $1$ .

$- 3000 \frac{1 + t^{2} - 2 (t^{1} t^{1})}{{(1 + t^{2})}^{2}}$

Step 6

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

$- 3000 \frac{1 + t^{2} - 2 t^{1 + 1}}{{(1 + t^{2})}^{2}}$

Step 7

Add $1$ and $1$ .

$- 3000 \frac{1 + t^{2} - 2 t^{2}}{{(1 + t^{2})}^{2}}$

Step 8

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

$- 3000 \frac{1 - t^{2}}{{(1 + t^{2})}^{2}}$

Step 9

Combine $- 3000$ and $\frac{1 - t^{2}}{{(1 + t^{2})}^{2}}$ .

$\frac{- 3000 (1 - t^{2})}{{(1 + t^{2})}^{2}}$

Step 10

Move the negative in front of the fraction.

$- \frac{3000 (1 - t^{2})}{{(1 + t^{2})}^{2}}$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$- \frac{3000 \cdot 1 + 3000 (- t^{2})}{{(1 + t^{2})}^{2}}$

Step 11.2

Simplify each term.

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Step 11.2.1

Multiply $3000$ by $1$ .

$- \frac{3000 + 3000 (- t^{2})}{{(1 + t^{2})}^{2}}$

Step 11.2.2

Multiply $- 1$ by $3000$ .

$- \frac{3000 - 3000 t^{2}}{{(1 + t^{2})}^{2}}$