Calculus Examples

$g (t) = - 3 \sqrt{7 t^{3 - 1}}$

Step 1

Simplify the expression.

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Step 1.1

Subtract $1$ from $3$ .

$\frac{d}{d t} [- 3 \sqrt{7 t^{2}}]$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 t^{2}}$ as ${(7 t^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d t} [- 3 {(7 t^{2})}^{\frac{1}{2}}]$

Step 2

Factor $7$ out of $7 t^{2}$ .

$\frac{d}{d t} [- 3 {(7 (t^{2}))}^{\frac{1}{2}}]$

Step 3

Apply basic rules of exponents.

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Step 3.1

Apply the product rule to $7 (t^{2})$ .

$\frac{d}{d t} [- 3 (7^{\frac{1}{2}} {(t^{2})}^{\frac{1}{2}})]$

Step 3.2

Multiply the exponents in ${(t^{2})}^{\frac{1}{2}}$ .

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Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d t} [- 3 (7^{\frac{1}{2}} t^{2 (\frac{1}{2})})]$

Step 3.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{d}{d t} [- 3 (7^{\frac{1}{2}} t^{2 (\frac{1}{2})})]$

Step 3.2.2.2

Rewrite the expression.

$\frac{d}{d t} [- 3 (7^{\frac{1}{2}} t^{1})]$

Step 4

Simplify.

$\frac{d}{d t} [- 3 (7^{\frac{1}{2}} t)]$

Step 5

Since $- 3 \cdot 7^{\frac{1}{2}}$ is constant with respect to $t$ , the derivative of $- 3 \cdot 7^{\frac{1}{2}} t$ with respect to $t$ is $- 3 \cdot 7^{\frac{1}{2}} \frac{d}{d t} [t]$ .

$- 3 \cdot 7^{\frac{1}{2}} \frac{d}{d t} [t]$

Step 6

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

$- 3 \cdot 7^{\frac{1}{2}} \cdot 1$

Step 7

Multiply $- 3$ by $1$ .

$- 3 \cdot 7^{\frac{1}{2}}$