Calculus Examples

$\frac{\sqrt{5}}{t^{3}}$

Step 1

Since $\sqrt{5}$ is constant with respect to $t$ , the derivative of $\frac{\sqrt{5}}{t^{3}}$ with respect to $t$ is $\sqrt{5} \frac{d}{d t} [\frac{1}{t^{3}}]$ .

$\sqrt{5} \frac{d}{d t} [\frac{1}{t^{3}}]$

Step 2

Apply basic rules of exponents.

Tap for more steps...

Step 2.1

Rewrite $\frac{1}{t^{3}}$ as ${(t^{3})}^{- 1}$ .

$\sqrt{5} \frac{d}{d t} [{(t^{3})}^{- 1}]$

Step 2.2

Multiply the exponents in ${(t^{3})}^{- 1}$ .

Tap for more steps...

Step 2.2.1

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

$\sqrt{5} \frac{d}{d t} [t^{3 \cdot - 1}]$

Step 2.2.2

Multiply $3$ by $- 1$ .

$\sqrt{5} \frac{d}{d t} [t^{- 3}]$

Step 3

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

$\sqrt{5} (- 3 t^{- 4})$

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

$\sqrt{5} (- 3 \frac{1}{t^{4}})$

Step 4.2

Combine terms.

Tap for more steps...

Step 4.2.1

Combine $- 3$ and $\frac{1}{t^{4}}$ .

$\sqrt{5} \frac{- 3}{t^{4}}$

Step 4.2.2

Move the negative in front of the fraction.

$\sqrt{5} (- \frac{3}{t^{4}})$

Step 4.2.3

Combine $\sqrt{5}$ and $\frac{3}{t^{4}}$ .

$- \frac{\sqrt{5} \cdot 3}{t^{4}}$

Step 4.2.4

Move $3$ to the left of $\sqrt{5}$ .

$- \frac{3 \sqrt{5}}{t^{4}}$