Calculus Examples

$v = i r$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d r'} r' = \frac{d}{d r'} (i r)$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $i$ is constant with respect to $r'$ , the derivative of $i r$ with respect to $r'$ is $i \frac{d}{d v} [r]$ .

$i \frac{d}{d v} [r]$

Step 3.2

Rewrite $\frac{d}{d v} [r]$ as $r'$ .

$i r'$

Step 4

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

$1 = i r'$

Step 5

Solve for $r'$ .

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Step 5.1

Rewrite the equation as $i r' = 1$ .

$i r' = 1$

Step 5.2

Divide each term in $i r' = 1$ by $i$ and simplify.

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Step 5.2.1

Divide each term in $i r' = 1$ by $i$ .

$\frac{i r'}{i} = \frac{1}{i}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $i$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{i r'}{i} = \frac{1}{i}$

Step 5.2.2.1.2

Divide $r'$ by $1$ .

$r' = \frac{1}{i}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply the numerator and denominator of $\frac{1}{i}$ by the conjugate of $i$ to make the denominator real.

$r' = \frac{1}{i} \cdot \frac{i}{i}$

Step 5.2.3.2

Multiply.

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Step 5.2.3.2.1

Combine.

$r' = \frac{1 i}{i i}$

Step 5.2.3.2.2

Multiply $i$ by $1$ .

$r' = \frac{i}{i i}$

Step 5.2.3.2.3

Simplify the denominator.

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Step 5.2.3.2.3.1

Raise $i$ to the power of $1$ .

$r' = \frac{i}{i^{1} i}$

Step 5.2.3.2.3.2

Raise $i$ to the power of $1$ .

$r' = \frac{i}{i^{1} i^{1}}$

Step 5.2.3.2.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r' = \frac{i}{i^{1 + 1}}$

Step 5.2.3.2.3.4

Add $1$ and $1$ .

$r' = \frac{i}{i^{2}}$

Step 5.2.3.2.3.5

Rewrite $i^{2}$ as $- 1$ .

$r' = \frac{i}{- 1}$

Step 5.2.3.3

Move the negative one from the denominator of $\frac{i}{- 1}$ .

$r' = - 1 \cdot i$

Step 5.2.3.4

Rewrite $- 1 \cdot i$ as $- i$ .

$r' = - i$

Step 6

Replace $r'$ with $\frac{d r}{d v}$ .

$\frac{d r}{d v} = - i$