Algebra Examples

$P = \frac{V^{2} R}{{(R + r)}^{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d r} (P) = \frac{d}{d r} (\frac{V^{2} R}{{(R + r)}^{2}})$

Step 2

The derivative of $P$ with respect to $r$ is $P'$ .

$P'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Since $V^{2} R$ is constant with respect to $r$ , the derivative of $\frac{V^{2} R}{{(R + r)}^{2}}$ with respect to $r$ is $V^{2} R \frac{d}{d r} [\frac{1}{{(R + r)}^{2}}]$ .

$V^{2} R \frac{d}{d r} [\frac{1}{{(R + r)}^{2}}]$

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

Rewrite $\frac{1}{{(R + r)}^{2}}$ as ${({(R + r)}^{2})}^{- 1}$ .

$V^{2} R \frac{d}{d r} [{({(R + r)}^{2})}^{- 1}]$

Step 3.1.2.2

Multiply the exponents in ${({(R + r)}^{2})}^{- 1}$ .

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Step 3.1.2.2.1

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

$V^{2} R \frac{d}{d r} [{(R + r)}^{2 \cdot - 1}]$

Step 3.1.2.2.2

Multiply $2$ by $- 1$ .

$V^{2} R \frac{d}{d r} [{(R + r)}^{- 2}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d r} [f (g (r))]$ is $f' (g (r)) g' (r)$ where $f (r) = r^{- 2}$ and $g (r) = R + r$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $R + r$ .

$V^{2} R (\frac{d}{d u} [u^{- 2}] \frac{d}{d r} [R + r])$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 2$ .

$V^{2} R (- 2 u^{- 3} \frac{d}{d r} [R + r])$

Step 3.2.3

Replace all occurrences of $u$ with $R + r$ .

$V^{2} R (- 2 {(R + r)}^{- 3} \frac{d}{d r} [R + r])$

Step 3.3

Differentiate.

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Step 3.3.1

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

$V^{2} R (- 2 {(R + r)}^{- 3} (\frac{d}{d r} [R] + \frac{d}{d r} [r]))$

Step 3.3.2

Since $R$ is constant with respect to $r$ , the derivative of $R$ with respect to $r$ is $0$ .

$V^{2} R (- 2 {(R + r)}^{- 3} (0 + \frac{d}{d r} [r]))$

Step 3.3.3

Add $0$ and $\frac{d}{d r} [r]$ .

$V^{2} R (- 2 {(R + r)}^{- 3} \frac{d}{d r} [r])$

Step 3.3.4

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

$V^{2} R (- 2 {(R + r)}^{- 3} \cdot 1)$

Step 3.3.5

Multiply $- 2$ by $1$ .

$V^{2} R (- 2 {(R + r)}^{- 3})$

Step 3.4

Simplify.

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Step 3.4.1

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

$V^{2} R (- 2 \frac{1}{{(R + r)}^{3}})$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $- 2$ and $\frac{1}{{(R + r)}^{3}}$ .

$V^{2} R \frac{- 2}{{(R + r)}^{3}}$

Step 3.4.2.2

Move the negative in front of the fraction.

$V^{2} R (- \frac{2}{{(R + r)}^{3}})$

Step 3.4.2.3

Combine $V^{2}$ and $\frac{2}{{(R + r)}^{3}}$ .

$R (- \frac{V^{2} \cdot 2}{{(R + r)}^{3}})$

Step 3.4.2.4

Combine $R$ and $\frac{V^{2} \cdot 2}{{(R + r)}^{3}}$ .

$- \frac{R (V^{2} \cdot 2)}{{(R + r)}^{3}}$

Step 3.4.2.5

Move $2$ to the left of $V^{2}$ .

$- \frac{R (2 \cdot V^{2})}{{(R + r)}^{3}}$

Step 3.4.2.6

Move $2$ to the left of $R$ .

$- \frac{2 R V^{2}}{{(R + r)}^{3}}$

Step 4

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

$P' = - \frac{2 R V^{2}}{{(R + r)}^{3}}$

Step 5

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

$\frac{d P}{d r} = - \frac{2 R V^{2}}{{(R + r)}^{3}}$