Calculus Examples

$(L + m) (V + n) = k$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d m} ((L + m) (V + n)) = \frac{d}{d m} (k)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Since $V + n$ is constant with respect to $m$ , the derivative of $(L + m) (V + n)$ with respect to $m$ is $(V + n) \frac{d}{d m} [L + m]$ .

$(V + n) \frac{d}{d m} [L + m]$

Step 2.2

By the Sum Rule, the derivative of $L + m$ with respect to $m$ is $\frac{d}{d m} [L] + \frac{d}{d m} [m]$ .

$(V + n) (\frac{d}{d m} [L] + \frac{d}{d m} [m])$

Step 2.3

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

$(V + n) (L' + \frac{d}{d m} [m])$

Step 2.4

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

$(V + n) (L' + 1)$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$(V + n) L' + (V + n) \cdot 1$

Step 2.5.2

Apply the distributive property.

$V L' + n L' + (V + n) \cdot 1$

Step 2.5.3

Apply the distributive property.

$V L' + n L' + V \cdot 1 + n \cdot 1$

Step 2.5.4

Combine terms.

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Step 2.5.4.1

Multiply $V$ by $1$ .

$V L' + n L' + V + n \cdot 1$

Step 2.5.4.2

Multiply $n$ by $1$ .

$V L' + n L' + V + n$

Step 2.5.5

Reorder terms.

$V + V L' + n L' + n$

Step 3

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

$0$

Step 4

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

$V + V L' + n L' + n = 0$

Step 5

Solve for $L'$ .

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Step 5.1

Move all terms not containing $L'$ to the right side of the equation.

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Step 5.1.1

Subtract $V$ from both sides of the equation.

$V L' + n L' + n = - V$

Step 5.1.2

Subtract $n$ from both sides of the equation.

$V L' + n L' = - V - n$

Step 5.2

Factor $L'$ out of $V L' + n L'$ .

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Step 5.2.1

Factor $L'$ out of $V L'$ .

$L' V + n L' = - V - n$

Step 5.2.2

Factor $L'$ out of $n L'$ .

$L' V + L' n = - V - n$

Step 5.2.3

Factor $L'$ out of $L' V + L' n$ .

$L' (V + n) = - V - n$

Step 5.3

Divide each term in $L' (V + n) = - V - n$ by $V + n$ and simplify.

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Step 5.3.1

Divide each term in $L' (V + n) = - V - n$ by $V + n$ .

$\frac{L' (V + n)}{V + n} = \frac{- V}{V + n} + \frac{- n}{V + n}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $V + n$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{L' (V + n)}{V + n} = \frac{- V}{V + n} + \frac{- n}{V + n}$

Step 5.3.2.1.2

Divide $L'$ by $1$ .

$L' = \frac{- V}{V + n} + \frac{- n}{V + n}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Combine the numerators over the common denominator.

$L' = \frac{- V - n}{V + n}$

Step 5.3.3.2

Cancel the common factor of $- V - n$ and $V + n$ .

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Step 5.3.3.2.1

Factor $- 1$ out of $- V$ .

$L' = \frac{- (V) - n}{V + n}$

Step 5.3.3.2.2

Factor $- 1$ out of $- n$ .

$L' = \frac{- (V) - (n)}{V + n}$

Step 5.3.3.2.3

Factor $- 1$ out of $- (V) - (n)$ .

$L' = \frac{- (V + n)}{V + n}$

Step 5.3.3.2.4

Rewrite $- (V + n)$ as $- 1 (V + n)$ .

$L' = \frac{- 1 (V + n)}{V + n}$

Step 5.3.3.2.5

Cancel the common factor.

$L' = \frac{- 1 (V + n)}{V + n}$

Step 5.3.3.2.6

Divide $- 1$ by $1$ .

$L' = - 1$

Step 6

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

$\frac{d L}{d m} = - 1$