Calculus Examples

$s = \sqrt{t} + 7 t^{2}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$s = t^{\frac{1}{2}} + 7 t^{2}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (t^{\frac{1}{2}} + 7 t^{2})$

Step 3

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

$s'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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

Step 4.2

Evaluate $\frac{d}{d t} [t^{\frac{1}{2}}]$ .

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Step 4.2.1

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

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

Step 4.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.2.3

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 1$ by $2$ .

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

Step 4.2.5.2

Subtract $2$ from $1$ .

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

Step 4.2.6

Move the negative in front of the fraction.

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

Step 4.3

Evaluate $\frac{d}{d t} [7 t^{2}]$ .

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Multiply $2$ by $7$ .

$\frac{1}{2} t^{- \frac{1}{2}} + 14 t$

Step 4.4

Simplify.

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Step 4.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \cdot \frac{1}{t^{\frac{1}{2}}} + 14 t$

Step 4.4.2

Multiply $\frac{1}{2}$ by $\frac{1}{t^{\frac{1}{2}}}$ .

$\frac{1}{2 t^{\frac{1}{2}}} + 14 t$

Step 4.4.3

Reorder terms.

$14 t + \frac{1}{2 t^{\frac{1}{2}}}$

Step 5

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

$s' = 14 t + \frac{1}{2 t^{\frac{1}{2}}}$

Step 6

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

$\frac{d s}{d t} = 14 t + \frac{1}{2 t^{\frac{1}{2}}}$