Calculus Examples

$4 t^{3} - \sqrt{t} + 2$

Step 1

By the Sum Rule, the derivative of $4 t^{3} - \sqrt{t} + 2$ with respect to $t$ is $\frac{d}{d t} [4 t^{3}] + \frac{d}{d t} [- \sqrt{t}] + \frac{d}{d t} [2]$ .

$\frac{d}{d t} [4 t^{3}] + \frac{d}{d t} [- \sqrt{t}] + \frac{d}{d t} [2]$

Step 2

Evaluate $\frac{d}{d t} [4 t^{3}]$ .

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Step 2.1

Since $4$ is constant with respect to $t$ , the derivative of $4 t^{3}$ with respect to $t$ is $4 \frac{d}{d t} [t^{3}]$ .

$4 \frac{d}{d t} [t^{3}] + \frac{d}{d t} [- \sqrt{t}] + \frac{d}{d t} [2]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 3$ .

$4 (3 t^{2}) + \frac{d}{d t} [- \sqrt{t}] + \frac{d}{d t} [2]$

Step 2.3

Multiply $3$ by $4$ .

$12 t^{2} + \frac{d}{d t} [- \sqrt{t}] + \frac{d}{d t} [2]$

Step 3

Evaluate $\frac{d}{d t} [- \sqrt{t}]$ .

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Step 3.1

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

$12 t^{2} + \frac{d}{d t} [- t^{\frac{1}{2}}] + \frac{d}{d t} [2]$

Step 3.2

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

$12 t^{2} - \frac{d}{d t} [t^{\frac{1}{2}}] + \frac{d}{d t} [2]$

Step 3.3

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

$12 t^{2} - (\frac{1}{2} t^{\frac{1}{2} - 1}) + \frac{d}{d t} [2]$

Step 3.4

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

$12 t^{2} - (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d t} [2]$

Step 3.5

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

$12 t^{2} - (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d t} [2]$

Step 3.6

Combine the numerators over the common denominator.

$12 t^{2} - (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d t} [2]$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

$12 t^{2} - (\frac{1}{2} t^{\frac{1 - 2}{2}}) + \frac{d}{d t} [2]$

Step 3.7.2

Subtract $2$ from $1$ .

$12 t^{2} - (\frac{1}{2} t^{\frac{- 1}{2}}) + \frac{d}{d t} [2]$

Step 3.8

Move the negative in front of the fraction.

$12 t^{2} - (\frac{1}{2} t^{- \frac{1}{2}}) + \frac{d}{d t} [2]$

Step 3.9

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

$12 t^{2} - \frac{t^{- \frac{1}{2}}}{2} + \frac{d}{d t} [2]$

Step 3.10

Move $t^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$12 t^{2} - \frac{1}{2 t^{\frac{1}{2}}} + \frac{d}{d t} [2]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $2$ is constant with respect to $t$ , the derivative of $2$ with respect to $t$ is $0$ .

$12 t^{2} - \frac{1}{2 t^{\frac{1}{2}}} + 0$

Step 4.2

Add $12 t^{2} - \frac{1}{2 t^{\frac{1}{2}}}$ and $0$ .

$12 t^{2} - \frac{1}{2 t^{\frac{1}{2}}}$