Calculus Examples

$\frac{t}{7 + t^{2}}$

Step 1

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 $g (t) = 7 + t^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

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

$\frac{(7 + t^{2}) \cdot 1 - t \frac{d}{d t} [7 + t^{2}]}{{(7 + t^{2})}^{2}}$

Step 2.2

Multiply $7 + t^{2}$ by $1$ .

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

Step 2.3

By the Sum Rule, the derivative of $7 + t^{2}$ with respect to $t$ is $\frac{d}{d t} [7] + \frac{d}{d t} [t^{2}]$ .

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

Step 2.4

Since $7$ is constant with respect to $t$ , the derivative of $7$ with respect to $t$ is $0$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Multiply $2$ by $- 1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

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

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

Step 8

Reorder terms.

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