Calculus Examples

$y = {(\frac{x - 5}{2 x + 1})}^{3}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \frac{x - 5}{2 x + 1}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x - 5}{2 x + 1}$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [\frac{x - 5}{2 x + 1}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2} \frac{d}{d x} [\frac{x - 5}{2 x + 1}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x - 5}{2 x + 1}$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{d}{d x} [\frac{x - 5}{2 x + 1}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x - 5$ and $g (x) = 2 x + 1$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{(2 x + 1) \frac{d}{d x} [x - 5] - (x - 5) \frac{d}{d x} [2 x + 1]}{{(2 x + 1)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{(2 x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 5]) - (x - 5) \frac{d}{d x} [2 x + 1]}{{(2 x + 1)}^{2}}$

Step 3.2

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

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{(2 x + 1) (1 + \frac{d}{d x} [- 5]) - (x - 5) \frac{d}{d x} [2 x + 1]}{{(2 x + 1)}^{2}}$

Step 3.3

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{(2 x + 1) (1 + 0) - (x - 5) \frac{d}{d x} [2 x + 1]}{{(2 x + 1)}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{(2 x + 1) \cdot 1 - (x - 5) \frac{d}{d x} [2 x + 1]}{{(2 x + 1)}^{2}}$

Step 3.4.2

Multiply $2 x + 1$ by $1$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{2 x + 1 - (x - 5) \frac{d}{d x} [2 x + 1]}{{(2 x + 1)}^{2}}$

Step 3.5

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{2 x + 1 - (x - 5) (\frac{d}{d x} [2 x] + \frac{d}{d x} [1])}{{(2 x + 1)}^{2}}$

Step 3.6

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{2 x + 1 - (x - 5) (2 \frac{d}{d x} [x] + \frac{d}{d x} [1])}{{(2 x + 1)}^{2}}$

Step 3.7

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

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{2 x + 1 - (x - 5) (2 \cdot 1 + \frac{d}{d x} [1])}{{(2 x + 1)}^{2}}$

Step 3.8

Multiply $2$ by $1$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{2 x + 1 - (x - 5) (2 + \frac{d}{d x} [1])}{{(2 x + 1)}^{2}}$

Step 3.9

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

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{2 x + 1 - (x - 5) (2 + 0)}{{(2 x + 1)}^{2}}$

Step 3.10

Simplify the expression.

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Step 3.10.1

Add $2$ and $0$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{2 x + 1 - (x - 5) \cdot 2}{{(2 x + 1)}^{2}}$

Step 3.10.2

Multiply $2$ by $- 1$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{2 x + 1 - 2 (x - 5)}{{(2 x + 1)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{2 x + 1 - 2 x - 2 \cdot - 5}{{(2 x + 1)}^{2}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Combine the opposite terms in $2 x + 1 - 2 x - 2 \cdot - 5$ .

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Step 4.2.1.1

Subtract $2 x$ from $2 x$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{0 + 1 - 2 \cdot - 5}{{(2 x + 1)}^{2}}$

Step 4.2.1.2

Add $0$ and $1$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{1 - 2 \cdot - 5}{{(2 x + 1)}^{2}}$

Step 4.2.2

Multiply $- 2$ by $- 5$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{1 + 10}{{(2 x + 1)}^{2}}$

Step 4.2.3

Add $1$ and $10$ .

$3 {(\frac{x - 5}{2 x + 1})}^{2} \frac{11}{{(2 x + 1)}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the product rule to $\frac{x - 5}{2 x + 1}$ .

$3 \frac{{(x - 5)}^{2}}{{(2 x + 1)}^{2}} \frac{11}{{(2 x + 1)}^{2}}$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $3$ and $\frac{{(x - 5)}^{2}}{{(2 x + 1)}^{2}}$ .

$\frac{3 {(x - 5)}^{2}}{{(2 x + 1)}^{2}} \cdot \frac{11}{{(2 x + 1)}^{2}}$

Step 5.2.2

Multiply $\frac{3 {(x - 5)}^{2}}{{(2 x + 1)}^{2}}$ by $\frac{11}{{(2 x + 1)}^{2}}$ .

$\frac{3 {(x - 5)}^{2} \cdot 11}{{(2 x + 1)}^{2} {(2 x + 1)}^{2}}$

Step 5.2.3

Multiply $11$ by $3$ .

$\frac{33 {(x - 5)}^{2}}{{(2 x + 1)}^{2} {(2 x + 1)}^{2}}$

Step 5.2.4

Multiply ${(2 x + 1)}^{2}$ by ${(2 x + 1)}^{2}$ by adding the exponents.

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Step 5.2.4.1

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

$\frac{33 {(x - 5)}^{2}}{{(2 x + 1)}^{2 + 2}}$

Step 5.2.4.2

Add $2$ and $2$ .

$\frac{33 {(x - 5)}^{2}}{{(2 x + 1)}^{4}}$