Calculus Examples

$x = \frac{t^{2} + 1}{3 t}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (x) = \frac{d}{d t} (\frac{t^{2} + 1}{3 t})$

Step 2

The derivative of $x$ with respect to $t$ is $x'$ .

$x'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

$\frac{1}{3} \cdot \frac{t (2 t + 0) - (t^{2} + 1) \frac{d}{d t} [t]}{t^{2}}$

Step 3.3.4

Add $2 t$ and $0$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Add $1$ and $1$ .

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

Step 3.8

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

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

Step 3.9

Combine fractions.

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Step 3.9.1

Multiply $- 1$ by $1$ .

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

Step 3.9.2

Multiply $\frac{1}{3}$ by $\frac{2 t^{2} - (t^{2} + 1)}{t^{2}}$ .

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

Step 3.10

Simplify.

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Step 3.10.1

Apply the distributive property.

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

Step 3.10.2

Simplify the numerator.

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Step 3.10.2.1

Multiply $- 1$ by $1$ .

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

Step 3.10.2.2

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

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

Step 3.10.3

Simplify the numerator.

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Step 3.10.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.10.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = t$ and $b = 1$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

$x' = \frac{(t + 1) (t - 1)}{3 t^{2}}$

Step 5

Replace $x'$ with $\frac{d x}{d t}$ .

$\frac{d x}{d t} = \frac{(t + 1) (t - 1)}{3 t^{2}}$