Calculus Examples

$y = \frac{27 R}{15 R + 54}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Cancel the common factor of $27$ and $15 R + 54$ .

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Step 1.1.1

Factor $3$ out of $27 R$ .

$\frac{d}{d R} [\frac{3 (9 R)}{15 R + 54}]$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $3$ out of $15 R$ .

$\frac{d}{d R} [\frac{3 (9 R)}{3 (5 R) + 54}]$

Step 1.1.2.2

Factor $3$ out of $54$ .

$\frac{d}{d R} [\frac{3 (9 R)}{3 (5 R) + 3 (18)}]$

Step 1.1.2.3

Factor $3$ out of $3 (5 R) + 3 (18)$ .

$\frac{d}{d R} [\frac{3 (9 R)}{3 (5 R + 18)}]$

Step 1.1.2.4

Cancel the common factor.

$\frac{d}{d R} [\frac{3 (9 R)}{3 (5 R + 18)}]$

Step 1.1.2.5

Rewrite the expression.

$\frac{d}{d R} [\frac{9 R}{5 R + 18}]$

Step 1.2

Since $9$ is constant with respect to $R$ , the derivative of $\frac{9 R}{5 R + 18}$ with respect to $R$ is $9 \frac{d}{d R} [\frac{R}{5 R + 18}]$ .

$9 \frac{d}{d R} [\frac{R}{5 R + 18}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d R} [\frac{f (R)}{g (R)}]$ is $\frac{g (R) \frac{d}{d R} [f (R)] - f (R) \frac{d}{d R} [g (R)]}{{g (R)}^{2}}$ where $f (R) = R$ and $g (R) = 5 R + 18$ .

$9 \frac{(5 R + 18) \frac{d}{d R} [R] - R \frac{d}{d R} [5 R + 18]}{{(5 R + 18)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$9 \frac{(5 R + 18) \cdot 1 - R \frac{d}{d R} [5 R + 18]}{{(5 R + 18)}^{2}}$

Step 3.2

Multiply $5 R + 18$ by $1$ .

$9 \frac{5 R + 18 - R \frac{d}{d R} [5 R + 18]}{{(5 R + 18)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $5 R + 18$ with respect to $R$ is $\frac{d}{d R} [5 R] + \frac{d}{d R} [18]$ .

$9 \frac{5 R + 18 - R (\frac{d}{d R} [5 R] + \frac{d}{d R} [18])}{{(5 R + 18)}^{2}}$

Step 3.4

Since $5$ is constant with respect to $R$ , the derivative of $5 R$ with respect to $R$ is $5 \frac{d}{d R} [R]$ .

$9 \frac{5 R + 18 - R (5 \frac{d}{d R} [R] + \frac{d}{d R} [18])}{{(5 R + 18)}^{2}}$

Step 3.5

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

$9 \frac{5 R + 18 - R (5 \cdot 1 + \frac{d}{d R} [18])}{{(5 R + 18)}^{2}}$

Step 3.6

Multiply $5$ by $1$ .

$9 \frac{5 R + 18 - R (5 + \frac{d}{d R} [18])}{{(5 R + 18)}^{2}}$

Step 3.7

Since $18$ is constant with respect to $R$ , the derivative of $18$ with respect to $R$ is $0$ .

$9 \frac{5 R + 18 - R (5 + 0)}{{(5 R + 18)}^{2}}$

Step 3.8

Simplify terms.

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Step 3.8.1

Add $5$ and $0$ .

$9 \frac{5 R + 18 - R \cdot 5}{{(5 R + 18)}^{2}}$

Step 3.8.2

Multiply $5$ by $- 1$ .

$9 \frac{5 R + 18 - 5 R}{{(5 R + 18)}^{2}}$

Step 3.8.3

Subtract $5 R$ from $5 R$ .

$9 \frac{0 + 18}{{(5 R + 18)}^{2}}$

Step 3.8.4

Add $0$ and $18$ .

$9 \frac{18}{{(5 R + 18)}^{2}}$

Step 3.8.5

Combine $9$ and $\frac{18}{{(5 R + 18)}^{2}}$ .

$\frac{9 \cdot 18}{{(5 R + 18)}^{2}}$

Step 3.8.6

Multiply $9$ by $18$ .

$\frac{162}{{(5 R + 18)}^{2}}$