Calculus Examples

$g (t) = \frac{2 t}{3 t - 1}$

Step 1

Since $2$ is constant with respect to $t$ , the derivative of $\frac{2 t}{3 t - 1}$ with respect to $t$ is $2 \frac{d}{d t} [\frac{t}{3 t - 1}]$ .

$2 \frac{d}{d t} [\frac{t}{3 t - 1}]$

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 $f (t) = t$ and $g (t) = 3 t - 1$ .

$2 \frac{(3 t - 1) \frac{d}{d t} [t] - t \frac{d}{d t} [3 t - 1]}{{(3 t - 1)}^{2}}$

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 $n = 1$ .

$2 \frac{(3 t - 1) \cdot 1 - t \frac{d}{d t} [3 t - 1]}{{(3 t - 1)}^{2}}$

Step 3.2

Multiply $3 t - 1$ by $1$ .

$2 \frac{3 t - 1 - t \frac{d}{d t} [3 t - 1]}{{(3 t - 1)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $3 t - 1$ with respect to $t$ is $\frac{d}{d t} [3 t] + \frac{d}{d t} [- 1]$ .

$2 \frac{3 t - 1 - t (\frac{d}{d t} [3 t] + \frac{d}{d t} [- 1])}{{(3 t - 1)}^{2}}$

Step 3.4

Since $3$ is constant with respect to $t$ , the derivative of $3 t$ with respect to $t$ is $3 \frac{d}{d t} [t]$ .

$2 \frac{3 t - 1 - t (3 \frac{d}{d t} [t] + \frac{d}{d t} [- 1])}{{(3 t - 1)}^{2}}$

Step 3.5

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

$2 \frac{3 t - 1 - t (3 \cdot 1 + \frac{d}{d t} [- 1])}{{(3 t - 1)}^{2}}$

Step 3.6

Multiply $3$ by $1$ .

$2 \frac{3 t - 1 - t (3 + \frac{d}{d t} [- 1])}{{(3 t - 1)}^{2}}$

Step 3.7

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

$2 \frac{3 t - 1 - t (3 + 0)}{{(3 t - 1)}^{2}}$

Step 3.8

Simplify terms.

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Step 3.8.1

Add $3$ and $0$ .

$2 \frac{3 t - 1 - t \cdot 3}{{(3 t - 1)}^{2}}$

Step 3.8.2

Multiply $3$ by $- 1$ .

$2 \frac{3 t - 1 - 3 t}{{(3 t - 1)}^{2}}$

Step 3.8.3

Subtract $3 t$ from $3 t$ .

$2 \frac{0 - 1}{{(3 t - 1)}^{2}}$

Step 3.8.4

Simplify the expression.

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Step 3.8.4.1

Subtract $1$ from $0$ .

$2 \frac{- 1}{{(3 t - 1)}^{2}}$

Step 3.8.4.2

Move the negative in front of the fraction.

$2 (- \frac{1}{{(3 t - 1)}^{2}})$

Step 3.8.4.3

Multiply $- 1$ by $2$ .

$- 2 \frac{1}{{(3 t - 1)}^{2}}$

Step 3.8.5

Combine $- 2$ and $\frac{1}{{(3 t - 1)}^{2}}$ .

$\frac{- 2}{{(3 t - 1)}^{2}}$

Step 3.8.6

Move the negative in front of the fraction.

$- \frac{2}{{(3 t - 1)}^{2}}$