Calculus Examples

$y = {(5 - 2 x)}^{- 3} + \frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}$

Step 1

By the Sum Rule, the derivative of ${(5 - 2 x)}^{- 3} + \frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}$ with respect to $x$ is $\frac{d}{d x} [{(5 - 2 x)}^{- 3}] + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$ .

$\frac{d}{d x} [{(5 - 2 x)}^{- 3}] + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$

Step 2

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

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Step 2.1

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 $g (x) = 5 - 2 x$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $5 - 2 x$ .

$\frac{d}{d u_{1}} [{u_{1}}^{- 3}] \frac{d}{d x} [5 - 2 x] + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$

Step 2.1.2

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

$- 3 {u_{1}}^{- 4} \frac{d}{d x} [5 - 2 x] + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $5 - 2 x$ .

$- 3 {(5 - 2 x)}^{- 4} \frac{d}{d x} [5 - 2 x] + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$

Step 2.2

By the Sum Rule, the derivative of $5 - 2 x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 2 x]$ .

$- 3 {(5 - 2 x)}^{- 4} (\frac{d}{d x} [5] + \frac{d}{d x} [- 2 x]) + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$

Step 2.3

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

$- 3 {(5 - 2 x)}^{- 4} (0 + \frac{d}{d x} [- 2 x]) + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$

Step 2.4

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

$- 3 {(5 - 2 x)}^{- 4} (0 - 2 \frac{d}{d x} [x]) + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$

Step 2.5

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

$- 3 {(5 - 2 x)}^{- 4} (0 - 2 \cdot 1) + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$

Step 2.6

Multiply $- 2$ by $1$ .

$- 3 {(5 - 2 x)}^{- 4} (0 - 2) + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$

Step 2.7

Subtract $2$ from $0$ .

$- 3 {(5 - 2 x)}^{- 4} \cdot - 2 + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$

Step 2.8

Multiply $- 2$ by $- 3$ .

$6 {(5 - 2 x)}^{- 4} + \frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$

Step 3

Evaluate $\frac{d}{d x} [\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}]$ .

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Step 3.1

Since $\frac{1}{8}$ is constant with respect to $x$ , the derivative of $\frac{1}{8} \cdot {(\frac{2}{x} + 1)}^{4}$ with respect to $x$ is $\frac{1}{8} \frac{d}{d x} [{(\frac{2}{x} + 1)}^{4}]$ .

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} \frac{d}{d x} [{(\frac{2}{x} + 1)}^{4}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \frac{2}{x} + 1$ .

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Step 3.2.1

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

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d x} [\frac{2}{x} + 1])$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} (4 {u_{2}}^{3} \frac{d}{d x} [\frac{2}{x} + 1])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $\frac{2}{x} + 1$ .

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} (4 {(\frac{2}{x} + 1)}^{3} \frac{d}{d x} [\frac{2}{x} + 1])$

Step 3.3

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

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} (4 {(\frac{2}{x} + 1)}^{3} (\frac{d}{d x} [\frac{2}{x}] + \frac{d}{d x} [1]))$

Step 3.4

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x}]$ .

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} (4 {(\frac{2}{x} + 1)}^{3} (2 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [1]))$

Step 3.5

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} (4 {(\frac{2}{x} + 1)}^{3} (2 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [1]))$

Step 3.6

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

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} (4 {(\frac{2}{x} + 1)}^{3} (2 (- x^{- 2}) + \frac{d}{d x} [1]))$

Step 3.7

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

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} (4 {(\frac{2}{x} + 1)}^{3} (2 (- x^{- 2}) + 0))$

Step 3.8

Multiply $- 1$ by $2$ .

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} (4 {(\frac{2}{x} + 1)}^{3} (- 2 x^{- 2} + 0))$

Step 3.9

Add $- 2 x^{- 2}$ and $0$ .

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} (4 {(\frac{2}{x} + 1)}^{3} (- 2 x^{- 2}))$

Step 3.10

Multiply $- 2$ by $4$ .

$6 {(5 - 2 x)}^{- 4} + \frac{1}{8} (- 8 {(\frac{2}{x} + 1)}^{3} x^{- 2})$

Step 3.11

Combine $- 8$ and $\frac{1}{8}$ .

$6 {(5 - 2 x)}^{- 4} + \frac{- 8}{8} ({(\frac{2}{x} + 1)}^{3} x^{- 2})$

Step 3.12

Combine ${(\frac{2}{x} + 1)}^{3}$ and $\frac{- 8}{8}$ .

$6 {(5 - 2 x)}^{- 4} + \frac{{(\frac{2}{x} + 1)}^{3} \cdot - 8}{8} x^{- 2}$

Step 3.13

Combine $\frac{{(\frac{2}{x} + 1)}^{3} \cdot - 8}{8}$ and $x^{- 2}$ .

$6 {(5 - 2 x)}^{- 4} + \frac{{(\frac{2}{x} + 1)}^{3} \cdot - 8 x^{- 2}}{8}$

Step 3.14

Move $- 8$ to the left of ${(\frac{2}{x} + 1)}^{3}$ .

$6 {(5 - 2 x)}^{- 4} + \frac{- 8 \cdot {(\frac{2}{x} + 1)}^{3} x^{- 2}}{8}$

Step 3.15

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 {(5 - 2 x)}^{- 4} + \frac{- 8 {(\frac{2}{x} + 1)}^{3}}{8 x^{2}}$

Step 3.16

Cancel the common factor of $- 8$ and $8$ .

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Step 3.16.1

Factor $8$ out of $- 8 {(\frac{2}{x} + 1)}^{3}$ .

$6 {(5 - 2 x)}^{- 4} + \frac{8 (- {(\frac{2}{x} + 1)}^{3})}{8 x^{2}}$

Step 3.16.2

Cancel the common factors.

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Step 3.16.2.1

Factor $8$ out of $8 x^{2}$ .

$6 {(5 - 2 x)}^{- 4} + \frac{8 (- {(\frac{2}{x} + 1)}^{3})}{8 (x^{2})}$

Step 3.16.2.2

Cancel the common factor.

$6 {(5 - 2 x)}^{- 4} + \frac{8 (- {(\frac{2}{x} + 1)}^{3})}{8 x^{2}}$

Step 3.16.2.3

Rewrite the expression.

$6 {(5 - 2 x)}^{- 4} + \frac{- {(\frac{2}{x} + 1)}^{3}}{x^{2}}$

Step 3.17

Move the negative in front of the fraction.

$6 {(5 - 2 x)}^{- 4} - \frac{{(\frac{2}{x} + 1)}^{3}}{x^{2}}$

Step 4

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

$6 \frac{1}{{(5 - 2 x)}^{4}} - \frac{{(\frac{2}{x} + 1)}^{3}}{x^{2}}$

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

Combine $6$ and $\frac{1}{{(5 - 2 x)}^{4}}$ .

$\frac{6}{{(5 - 2 x)}^{4}} - \frac{{(\frac{2}{x} + 1)}^{3}}{x^{2}}$

Step 5.1.2

To write $\frac{6}{{(5 - 2 x)}^{4}}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{6}{{(5 - 2 x)}^{4}} \cdot \frac{x^{2}}{x^{2}} - \frac{{(\frac{2}{x} + 1)}^{3}}{x^{2}}$

Step 5.1.3

To write $- \frac{{(\frac{2}{x} + 1)}^{3}}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{{(5 - 2 x)}^{4}}{{(5 - 2 x)}^{4}}$ .

$\frac{6}{{(5 - 2 x)}^{4}} \cdot \frac{x^{2}}{x^{2}} - \frac{{(\frac{2}{x} + 1)}^{3}}{x^{2}} \cdot \frac{{(5 - 2 x)}^{4}}{{(5 - 2 x)}^{4}}$

Step 5.1.4

Write each expression with a common denominator of ${(5 - 2 x)}^{4} x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1.4.1

Multiply $\frac{6}{{(5 - 2 x)}^{4}}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{6 x^{2}}{{(5 - 2 x)}^{4} x^{2}} - \frac{{(\frac{2}{x} + 1)}^{3}}{x^{2}} \cdot \frac{{(5 - 2 x)}^{4}}{{(5 - 2 x)}^{4}}$

Step 5.1.4.2

Multiply $\frac{{(\frac{2}{x} + 1)}^{3}}{x^{2}}$ by $\frac{{(5 - 2 x)}^{4}}{{(5 - 2 x)}^{4}}$ .

$\frac{6 x^{2}}{{(5 - 2 x)}^{4} x^{2}} - \frac{{(\frac{2}{x} + 1)}^{3} {(5 - 2 x)}^{4}}{x^{2} {(5 - 2 x)}^{4}}$

Step 5.1.4.3

Reorder the factors of ${(5 - 2 x)}^{4} x^{2}$ .

$\frac{6 x^{2}}{x^{2} {(5 - 2 x)}^{4}} - \frac{{(\frac{2}{x} + 1)}^{3} {(5 - 2 x)}^{4}}{x^{2} {(5 - 2 x)}^{4}}$

Step 5.1.5

Combine the numerators over the common denominator.

$\frac{6 x^{2} - {(\frac{2}{x} + 1)}^{3} {(5 - 2 x)}^{4}}{x^{2} {(5 - 2 x)}^{4}}$

Step 5.2

Reorder terms.

$\frac{6 x^{2} - {(5 - 2 x)}^{4} {(\frac{2}{x} + 1)}^{3}}{x^{2} {(5 - 2 x)}^{4}}$