Calculus Examples

$m = \frac{11}{n^{7}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d m} (m) = \frac{d}{d m} (\frac{11}{n^{7}})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $11$ is constant with respect to $m$ , the derivative of $\frac{11}{n^{7}}$ with respect to $m$ is $11 \frac{d}{d m} [\frac{1}{n^{7}}]$ .

$11 \frac{d}{d m} [\frac{1}{n^{7}}]$

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d m} [\frac{f (m)}{g (m)}]$ is $\frac{g (m) \frac{d}{d m} [f (m)] - f (m) \frac{d}{d m} [g (m)]}{{g (m)}^{2}}$ where $f (m) = 1$ and $g (m) = n^{7}$ .

$11 \frac{n^{7} \frac{d}{d m} [1] - 1 \cdot 1 \frac{d}{d m} [n^{7}]}{{(n^{7})}^{2}}$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Multiply $- 1$ by $1$ .

$11 \frac{n^{7} \frac{d}{d m} [1] - \frac{d}{d m} [n^{7}]}{{(n^{7})}^{2}}$

Step 3.3.2

Multiply the exponents in ${(n^{7})}^{2}$ .

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Step 3.3.2.1

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

$11 \frac{n^{7} \frac{d}{d m} [1] - \frac{d}{d m} [n^{7}]}{n^{7 \cdot 2}}$

Step 3.3.2.2

Multiply $7$ by $2$ .

$11 \frac{n^{7} \frac{d}{d m} [1] - \frac{d}{d m} [n^{7}]}{n^{14}}$

Step 3.3.3

Since $1$ is constant with respect to $m$ , the derivative of $1$ with respect to $m$ is $0$ .

$11 \frac{n^{7} \cdot 0 - \frac{d}{d m} [n^{7}]}{n^{14}}$

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Multiply $n^{7}$ by $0$ .

$11 \frac{0 - \frac{d}{d m} [n^{7}]}{n^{14}}$

Step 3.3.4.2

Subtract $\frac{d}{d m} [n^{7}]$ from $0$ .

$11 \frac{- \frac{d}{d m} [n^{7}]}{n^{14}}$

Step 3.3.4.3

Move the negative in front of the fraction.

$11 (- \frac{\frac{d}{d m} [n^{7}]}{n^{14}})$

Step 3.3.4.4

Multiply $- 1$ by $11$ .

$- 11 \frac{\frac{d}{d m} [n^{7}]}{n^{14}}$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d m} [f (g (m))]$ is $f' (g (m)) g' (m)$ where $f (m) = m^{7}$ and $g (m) = n$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $n$ .

$- 11 \frac{\frac{d}{d u} [u^{7}] \frac{d}{d m} [n]}{n^{14}}$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

$- 11 \frac{7 u^{6} \frac{d}{d m} [n]}{n^{14}}$

Step 3.4.3

Replace all occurrences of $u$ with $n$ .

$- 11 \frac{7 n^{6} \frac{d}{d m} [n]}{n^{14}}$

Step 3.5

Cancel the common factor of $n^{6}$ and $n^{14}$ .

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Step 3.5.1

Factor $n^{6}$ out of $7 n^{6} \frac{d}{d m} [n]$ .

$- 11 \frac{n^{6} (7 \frac{d}{d m} [n])}{n^{14}}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $n^{6}$ out of $n^{14}$ .

$- 11 \frac{n^{6} (7 \frac{d}{d m} [n])}{n^{6} n^{8}}$

Step 3.5.2.2

Cancel the common factor.

$- 11 \frac{n^{6} (7 \frac{d}{d m} [n])}{n^{6} n^{8}}$

Step 3.5.2.3

Rewrite the expression.

$- 11 \frac{7 \frac{d}{d m} [n]}{n^{8}}$

Step 3.6

Rewrite $\frac{d}{d m} [n]$ as $n'$ .

$- 11 \frac{7 n'}{n^{8}}$

Step 3.7

Combine $- 11$ and $\frac{7 n'}{n^{8}}$ .

$\frac{- 11 (7 n')}{n^{8}}$

Step 3.8

Simplify the expression.

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Step 3.8.1

Multiply $7$ by $- 11$ .

$\frac{- 77 n'}{n^{8}}$

Step 3.8.2

Move the negative in front of the fraction.

$- \frac{77 n'}{n^{8}}$

Step 4

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

$1 = - \frac{77 n'}{n^{8}}$

Step 5

Solve for $n'$ .

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Step 5.1

Rewrite the equation as $- \frac{77 n'}{n^{8}} = 1$ .

$- \frac{77 n'}{n^{8}} = 1$

Step 5.2

Divide each term in $- \frac{77 n'}{n^{8}} = 1$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- \frac{77 n'}{n^{8}} = 1$ by $- 1$ .

$\frac{- \frac{77 n'}{n^{8}}}{- 1} = \frac{1}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{77 n'}{n^{8}}}{1} = \frac{1}{- 1}$

Step 5.2.2.2

Divide $\frac{77 n'}{n^{8}}$ by $1$ .

$\frac{77 n'}{n^{8}} = \frac{1}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $1$ by $- 1$ .

$\frac{77 n'}{n^{8}} = - 1$

Step 5.3

Multiply both sides by $n^{8}$ .

$\frac{77 n'}{n^{8}} n^{8} = - n^{8}$

Step 5.4

Simplify the left side.

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Step 5.4.1

Cancel the common factor of $n^{8}$ .

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Step 5.4.1.1

Cancel the common factor.

$\frac{77 n'}{n^{8}} n^{8} = - n^{8}$

Step 5.4.1.2

Rewrite the expression.

$77 n' = - n^{8}$

Step 5.5

Divide each term in $77 n' = - n^{8}$ by $77$ and simplify.

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Step 5.5.1

Divide each term in $77 n' = - n^{8}$ by $77$ .

$\frac{77 n'}{77} = \frac{- n^{8}}{77}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $77$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{77 n'}{77} = \frac{- n^{8}}{77}$

Step 5.5.2.1.2

Divide $n'$ by $1$ .

$n' = \frac{- n^{8}}{77}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

$n' = - \frac{n^{8}}{77}$

Step 6

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

$\frac{d n}{d m} = - \frac{n^{8}}{77}$