Calculus Examples

$y = {(\frac{6 x^{2}}{2 - x})}^{3}$

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 $g (x) = \frac{6 x^{2}}{2 - x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{6 x^{2}}{2 - x}$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [\frac{6 x^{2}}{2 - x}]$

Step 1.2

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

$3 u^{2} \frac{d}{d x} [\frac{6 x^{2}}{2 - x}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{6 x^{2}}{2 - x}$ .

$3 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{d}{d x} [\frac{6 x^{2}}{2 - x}]$

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 x^{2}}{2 - x}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{x^{2}}{2 - x}]$ .

$3 {(\frac{6 x^{2}}{2 - x})}^{2} (6 \frac{d}{d x} [\frac{x^{2}}{2 - x}])$

Step 2.2

Multiply $6$ by $3$ .

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{d}{d x} [\frac{x^{2}}{2 - x}]$

Step 3

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 $f (x) = x^{2}$ and $g (x) = 2 - x$ .

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{(2 - x) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [2 - x]}{{(2 - x)}^{2}}$

Step 4

Differentiate.

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Step 4.1

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

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{(2 - x) (2 x) - x^{2} \frac{d}{d x} [2 - x]}{{(2 - x)}^{2}}$

Step 4.2

Move $2$ to the left of $2 - x$ .

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{2 \cdot (2 - x) x - x^{2} \frac{d}{d x} [2 - x]}{{(2 - x)}^{2}}$

Step 4.3

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

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{2 (2 - x) x - x^{2} (\frac{d}{d x} [2] + \frac{d}{d x} [- x])}{{(2 - x)}^{2}}$

Step 4.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{2 (2 - x) x - x^{2} (0 + \frac{d}{d x} [- x])}{{(2 - x)}^{2}}$

Step 4.5

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

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{2 (2 - x) x - x^{2} \frac{d}{d x} [- x]}{{(2 - x)}^{2}}$

Step 4.6

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

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{2 (2 - x) x - x^{2} (- \frac{d}{d x} [x])}{{(2 - x)}^{2}}$

Step 4.7

Multiply.

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Step 4.7.1

Multiply $- 1$ by $- 1$ .

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{2 (2 - x) x + 1 x^{2} \frac{d}{d x} [x]}{{(2 - x)}^{2}}$

Step 4.7.2

Multiply $x^{2}$ by $1$ .

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{2 (2 - x) x + x^{2} \frac{d}{d x} [x]}{{(2 - x)}^{2}}$

Step 4.8

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

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{2 (2 - x) x + x^{2} \cdot 1}{{(2 - x)}^{2}}$

Step 4.9

Combine fractions.

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Step 4.9.1

Multiply $x^{2}$ by $1$ .

$18 {(\frac{6 x^{2}}{2 - x})}^{2} \frac{2 (2 - x) x + x^{2}}{{(2 - x)}^{2}}$

Step 4.9.2

Combine $\frac{2 (2 - x) x + x^{2}}{{(2 - x)}^{2}}$ and $18$ .

$\frac{(2 (2 - x) x + x^{2}) \cdot 18}{{(2 - x)}^{2}} {(\frac{6 x^{2}}{2 - x})}^{2}$

Step 4.9.3

Move $18$ to the left of $2 (2 - x) x + x^{2}$ .

$\frac{18 \cdot (2 (2 - x) x + x^{2})}{{(2 - x)}^{2}} {(\frac{6 x^{2}}{2 - x})}^{2}$

Step 5

Simplify.

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Step 5.1

Apply the product rule to $\frac{6 x^{2}}{2 - x}$ .

$\frac{18 (2 (2 - x) x + x^{2})}{{(2 - x)}^{2}} \cdot \frac{{(6 x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.2

Apply the product rule to $6 x^{2}$ .

$\frac{18 (2 (2 - x) x + x^{2})}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.3

Apply the distributive property.

$\frac{18 ((2 \cdot 2 + 2 (- x)) x + x^{2})}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.4

Apply the distributive property.

$\frac{18 (2 \cdot 2 x + 2 (- x) x + x^{2})}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.5

Apply the distributive property.

$\frac{18 (2 \cdot 2 x) + 18 (2 (- x) x) + 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.6

Combine terms.

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Step 5.6.1

Multiply $2$ by $2$ .

$\frac{18 (4 x) + 18 (2 (- x) x) + 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.6.2

Multiply $4$ by $18$ .

$\frac{72 x + 18 (2 (- x) x) + 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.6.3

Multiply $- 1$ by $2$ .

$\frac{72 x + 18 (- 2 x \cdot x) + 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.6.4

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

$\frac{72 x + 18 (- 2 (x^{1} x)) + 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.6.5

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

$\frac{72 x + 18 (- 2 (x^{1} x^{1})) + 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.6.6

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

$\frac{72 x + 18 (- 2 x^{1 + 1}) + 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.6.7

Add $1$ and $1$ .

$\frac{72 x + 18 (- 2 x^{2}) + 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.6.8

Multiply $- 2$ by $18$ .

$\frac{72 x - 36 x^{2} + 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.6.9

Add $- 36 x^{2}$ and $18 x^{2}$ .

$\frac{72 x - 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{6^{2} {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.6.10

Raise $6$ to the power of $2$ .

$\frac{72 x - 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{36 {(x^{2})}^{2}}{{(2 - x)}^{2}}$

Step 5.6.11

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 5.6.11.1

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

$\frac{72 x - 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{36 x^{2 \cdot 2}}{{(2 - x)}^{2}}$

Step 5.6.11.2

Multiply $2$ by $2$ .

$\frac{72 x - 18 x^{2}}{{(2 - x)}^{2}} \cdot \frac{36 x^{4}}{{(2 - x)}^{2}}$

Step 5.6.12

Multiply $\frac{72 x - 18 x^{2}}{{(2 - x)}^{2}}$ by $\frac{36 x^{4}}{{(2 - x)}^{2}}$ .

$\frac{(72 x - 18 x^{2}) (36 x^{4})}{{(2 - x)}^{2} {(2 - x)}^{2}}$

Step 5.6.13

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

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Step 5.6.13.1

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

$\frac{(72 x - 18 x^{2}) (36 x^{4})}{{(2 - x)}^{2 + 2}}$

Step 5.6.13.2

Add $2$ and $2$ .

$\frac{(72 x - 18 x^{2}) (36 x^{4})}{{(2 - x)}^{4}}$

Step 5.6.14

Move $36$ to the left of $72 x - 18 x^{2}$ .

$\frac{36 (72 x - 18 x^{2}) x^{4}}{{(2 - x)}^{4}}$

Step 5.7

Reorder terms.

$\frac{36 x^{4} (72 x - 18 x^{2})}{{(- x + 2)}^{4}}$

Step 5.8

Simplify the numerator.

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Step 5.8.1

Factor $18 x$ out of $72 x - 18 x^{2}$ .

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Step 5.8.1.1

Factor $18 x$ out of $72 x$ .

$\frac{36 x^{4} (18 x (4) - 18 x^{2})}{{(- x + 2)}^{4}}$

Step 5.8.1.2

Factor $18 x$ out of $- 18 x^{2}$ .

$\frac{36 x^{4} (18 x (4) + 18 x (- x))}{{(- x + 2)}^{4}}$

Step 5.8.1.3

Factor $18 x$ out of $18 x (4) + 18 x (- x)$ .

$\frac{36 x^{4} (18 x (4 - x))}{{(- x + 2)}^{4}}$

$\frac{36 x^{4} \cdot 18 x (4 - x)}{{(- x + 2)}^{4}}$

Step 5.8.2

Combine exponents.

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Step 5.8.2.1

Multiply $18$ by $36$ .

$\frac{648 x^{4} x (4 - x)}{{(- x + 2)}^{4}}$

Step 5.8.2.2

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

$\frac{648 (x^{1} x^{4}) (4 - x)}{{(- x + 2)}^{4}}$

Step 5.8.2.3

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

$\frac{648 x^{1 + 4} (4 - x)}{{(- x + 2)}^{4}}$

Step 5.8.2.4

Add $1$ and $4$ .

$\frac{648 x^{5} (4 - x)}{{(- x + 2)}^{4}}$