Calculus Examples

$f (x) = \frac{7 t}{t^{2} + 1}$

Step 1

Since $7$ is constant with respect to $t$ , the derivative of $\frac{7 t}{t^{2} + 1}$ with respect to $t$ is $7 \frac{d}{d t} [\frac{t}{t^{2} + 1}]$ .

$7 \frac{d}{d t} [\frac{t}{t^{2} + 1}]$

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

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

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 $n = 1$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

$7 \frac{t^{2} + 1 - t (2 t + 0)}{{(t^{2} + 1)}^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $2 t$ and $0$ .

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

Step 3.6.2

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply $- 1$ by $7$ .

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

Step 10.2.2

Multiply $7$ by $1$ .

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