Calculus Examples

$m = \frac{7}{n^{3}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d n} (m) = \frac{d}{d n} (\frac{7}{n^{3}})$

Step 2

The derivative of $m$ with respect to $n$ is $m'$ .

$m'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$7 \frac{d}{d n} [\frac{1}{n^{3}}]$

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $3$ by $- 1$ .

$7 \frac{d}{d n} [n^{- 3}]$

Step 3.3

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

$7 (- 3 n^{- 4})$

Step 3.4

Multiply $- 3$ by $7$ .

$- 21 n^{- 4}$

Step 3.5

Simplify.

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Step 3.5.1

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

$- 21 \frac{1}{n^{4}}$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Combine $- 21$ and $\frac{1}{n^{4}}$ .

$\frac{- 21}{n^{4}}$

Step 3.5.2.2

Move the negative in front of the fraction.

$- \frac{21}{n^{4}}$

Step 4

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

$m' = - \frac{21}{n^{4}}$

Step 5

Replace $m'$ with $\frac{d m}{d n}$ .

$\frac{d m}{d n} = - \frac{21}{n^{4}}$