Calculus Examples

$y = {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4}$

Step 1

Let $y = f (x)$ , take the natural logarithm of both sides $\ln (y) = \ln (f (x))$ .

$\ln (y) = \ln ({(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4})$

Step 2

Expand the right hand side.

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Step 2.1

Rewrite $\ln ({(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4})$ as $\ln ({(x^{2} + 2)}^{2}) + \ln ({(x^{4} + 4)}^{4})$ .

$\ln (y) = \ln ({(x^{2} + 2)}^{2}) + \ln ({(x^{4} + 4)}^{4})$

Step 2.2

Expand $\ln ({(x^{2} + 2)}^{2})$ by moving $2$ outside the logarithm.

$\ln (y) = 2 \ln (x^{2} + 2) + \ln ({(x^{4} + 4)}^{4})$

Step 2.3

Expand $\ln ({(x^{4} + 4)}^{4})$ by moving $4$ outside the logarithm.

$\ln (y) = 2 \ln (x^{2} + 2) + 4 \ln (x^{4} + 4)$

Step 3

Differentiate the expression using the chain rule, keeping in mind that $y$ is a function of $x$ .

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Step 3.1

Differentiate the left hand side $\ln (y)$ using the chain rule.

$\frac{y'}{y} = 2 \ln (x^{2} + 2) + 4 \ln (x^{4} + 4)$

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate $2 \ln (x^{2} + 2) + 4 \ln (x^{4} + 4)$ .

$\frac{y'}{y} = \frac{d}{d x} [2 \ln (x^{2} + 2) + 4 \ln (x^{4} + 4)]$

Step 3.2.2

By the Sum Rule, the derivative of $2 \ln (x^{2} + 2) + 4 \ln (x^{4} + 4)$ with respect to $x$ is $\frac{d}{d x} [2 \ln (x^{2} + 2)] + \frac{d}{d x} [4 \ln (x^{4} + 4)]$ .

$\frac{y'}{y} = \frac{d}{d x} [2 \ln (x^{2} + 2)] + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3

Evaluate $\frac{d}{d x} [2 \ln (x^{2} + 2)]$ .

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Step 3.2.3.1

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

$\frac{y'}{y} = 2 \frac{d}{d x} [\ln (x^{2} + 2)] + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3.2

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

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Step 3.2.3.2.1

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

$\frac{y'}{y} = 2 (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [x^{2} + 2]) + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3.2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{y'}{y} = 2 (\frac{1}{u_{1}} \frac{d}{d x} [x^{2} + 2]) + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3.2.3

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

$\frac{y'}{y} = 2 (\frac{1}{x^{2} + 2} \frac{d}{d x} [x^{2} + 2]) + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3.3

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

$\frac{y'}{y} = 2 (\frac{1}{x^{2} + 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2])) + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3.4

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

$\frac{y'}{y} = 2 (\frac{1}{x^{2} + 2} (2 x + \frac{d}{d x} [2])) + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3.5

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

$\frac{y'}{y} = 2 (\frac{1}{x^{2} + 2} (2 x + 0)) + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3.6

Add $2 x$ and $0$ .

$\frac{y'}{y} = 2 (\frac{1}{x^{2} + 2} (2 x)) + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3.7

Combine $2$ and $\frac{1}{x^{2} + 2}$ .

$\frac{y'}{y} = 2 (\frac{2}{x^{2} + 2} x) + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3.8

Combine $\frac{2}{x^{2} + 2}$ and $x$ .

$\frac{y'}{y} = 2 \frac{2 x}{x^{2} + 2} + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3.9

Combine $2$ and $\frac{2 x}{x^{2} + 2}$ .

$\frac{y'}{y} = \frac{2 (2 x)}{x^{2} + 2} + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.3.10

Multiply $2$ by $2$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + \frac{d}{d x} [4 \ln (x^{4} + 4)]$

Step 3.2.4

Evaluate $\frac{d}{d x} [4 \ln (x^{4} + 4)]$ .

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Step 3.2.4.1

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

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + 4 \frac{d}{d x} [\ln (x^{4} + 4)]$

Step 3.2.4.2

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

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Step 3.2.4.2.1

To apply the Chain Rule, set $u_{2}$ as $x^{4} + 4$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + 4 (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [x^{4} + 4])$

Step 3.2.4.2.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + 4 (\frac{1}{u_{2}} \frac{d}{d x} [x^{4} + 4])$

Step 3.2.4.2.3

Replace all occurrences of $u_{2}$ with $x^{4} + 4$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + 4 (\frac{1}{x^{4} + 4} \frac{d}{d x} [x^{4} + 4])$

Step 3.2.4.3

By the Sum Rule, the derivative of $x^{4} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [4]$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + 4 (\frac{1}{x^{4} + 4} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [4]))$

Step 3.2.4.4

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

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + 4 (\frac{1}{x^{4} + 4} (4 x^{3} + \frac{d}{d x} [4]))$

Step 3.2.4.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + 4 (\frac{1}{x^{4} + 4} (4 x^{3} + 0))$

Step 3.2.4.6

Add $4 x^{3}$ and $0$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + 4 (\frac{1}{x^{4} + 4} (4 x^{3}))$

Step 3.2.4.7

Combine $4$ and $\frac{1}{x^{4} + 4}$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + 4 (\frac{4}{x^{4} + 4} x^{3})$

Step 3.2.4.8

Combine $\frac{4}{x^{4} + 4}$ and $x^{3}$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + 4 \frac{4 x^{3}}{x^{4} + 4}$

Step 3.2.4.9

Combine $4$ and $\frac{4 x^{3}}{x^{4} + 4}$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + \frac{4 (4 x^{3})}{x^{4} + 4}$

Step 3.2.4.10

Multiply $4$ by $4$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} + \frac{16 x^{3}}{x^{4} + 4}$

Step 3.2.5

Combine terms.

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Step 3.2.5.1

To write $\frac{4 x}{x^{2} + 2}$ as a fraction with a common denominator, multiply by $\frac{x^{4} + 4}{x^{4} + 4}$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} \cdot \frac{x^{4} + 4}{x^{4} + 4} + \frac{16 x^{3}}{x^{4} + 4}$

Step 3.2.5.2

To write $\frac{16 x^{3}}{x^{4} + 4}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 2}{x^{2} + 2}$ .

$\frac{y'}{y} = \frac{4 x}{x^{2} + 2} \cdot \frac{x^{4} + 4}{x^{4} + 4} + \frac{16 x^{3}}{x^{4} + 4} \cdot \frac{x^{2} + 2}{x^{2} + 2}$

Step 3.2.5.3

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

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Step 3.2.5.3.1

Multiply $\frac{4 x}{x^{2} + 2}$ by $\frac{x^{4} + 4}{x^{4} + 4}$ .

$\frac{y'}{y} = \frac{4 x (x^{4} + 4)}{(x^{2} + 2) (x^{4} + 4)} + \frac{16 x^{3}}{x^{4} + 4} \cdot \frac{x^{2} + 2}{x^{2} + 2}$

Step 3.2.5.3.2

Multiply $\frac{16 x^{3}}{x^{4} + 4}$ by $\frac{x^{2} + 2}{x^{2} + 2}$ .

$\frac{y'}{y} = \frac{4 x (x^{4} + 4)}{(x^{2} + 2) (x^{4} + 4)} + \frac{16 x^{3} (x^{2} + 2)}{(x^{4} + 4) (x^{2} + 2)}$

Step 3.2.5.3.3

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

$\frac{y'}{y} = \frac{4 x (x^{4} + 4)}{(x^{2} + 2) (x^{4} + 4)} + \frac{16 x^{3} (x^{2} + 2)}{(x^{2} + 2) (x^{4} + 4)}$

Step 3.2.5.4

Combine the numerators over the common denominator.

$\frac{y'}{y} = \frac{4 x (x^{4} + 4) + 16 x^{3} (x^{2} + 2)}{(x^{2} + 2) (x^{4} + 4)}$

Step 4

Isolate $y'$ and substitute the original function for $y$ in the right hand side.

$y' = \frac{4 x (x^{4} + 4) + 16 x^{3} (x^{2} + 2)}{(x^{2} + 2) (x^{4} + 4)} ({(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4})$

Step 5

Simplify the right hand side.

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Step 5.1

Cancel the common factor of $(x^{2} + 2) (x^{4} + 4)$ .

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Step 5.1.1

Factor $(x^{2} + 2) (x^{4} + 4)$ out of ${(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4}$ .

$y' = \frac{4 x (x^{4} + 4) + 16 x^{3} (x^{2} + 2)}{(x^{2} + 2) (x^{4} + 4)} ((x^{2} + 2) (x^{4} + 4) ((x^{2} + 2) {(x^{4} + 4)}^{3}))$

Step 5.1.2

Cancel the common factor.

$y' = \frac{4 x (x^{4} + 4) + 16 x^{3} (x^{2} + 2)}{(x^{2} + 2) (x^{4} + 4)} ((x^{2} + 2) (x^{4} + 4) ((x^{2} + 2) {(x^{4} + 4)}^{3}))$

Step 5.1.3

Rewrite the expression.

$y' = (4 x (x^{4} + 4) + 16 x^{3} (x^{2} + 2)) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$y' = (4 x \cdot x^{4} + 4 x \cdot 4 + 16 x^{3} (x^{2} + 2)) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.2.2

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 5.2.2.1

Move $x^{4}$ .

$y' = (4 (x^{4} x) + 4 x \cdot 4 + 16 x^{3} (x^{2} + 2)) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.2.2.2

Multiply $x^{4}$ by $x$ .

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Step 5.2.2.2.1

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

$y' = (4 (x^{4} x^{1}) + 4 x \cdot 4 + 16 x^{3} (x^{2} + 2)) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.2.2.2.2

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

$y' = (4 x^{4 + 1} + 4 x \cdot 4 + 16 x^{3} (x^{2} + 2)) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.2.2.3

Add $4$ and $1$ .

$y' = (4 x^{5} + 4 x \cdot 4 + 16 x^{3} (x^{2} + 2)) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.2.3

Multiply $4$ by $4$ .

$y' = (4 x^{5} + 16 x + 16 x^{3} (x^{2} + 2)) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.2.4

Apply the distributive property.

$y' = (4 x^{5} + 16 x + 16 x^{3} x^{2} + 16 x^{3} \cdot 2) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.2.5

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 5.2.5.1

Move $x^{2}$ .

$y' = (4 x^{5} + 16 x + 16 (x^{2} x^{3}) + 16 x^{3} \cdot 2) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.2.5.2

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

$y' = (4 x^{5} + 16 x + 16 x^{2 + 3} + 16 x^{3} \cdot 2) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.2.5.3

Add $2$ and $3$ .

$y' = (4 x^{5} + 16 x + 16 x^{5} + 16 x^{3} \cdot 2) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.2.6

Multiply $2$ by $16$ .

$y' = (4 x^{5} + 16 x + 16 x^{5} + 32 x^{3}) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.3

Add $4 x^{5}$ and $16 x^{5}$ .

$y' = (20 x^{5} + 16 x + 32 x^{3}) ((x^{2} + 2) {(x^{4} + 4)}^{3})$

Step 5.4

Apply the distributive property.

$y' = (20 x^{5} + 16 x + 32 x^{3}) (x^{2} {(x^{4} + 4)}^{3} + 2 {(x^{4} + 4)}^{3})$

Step 5.5

Expand $(20 x^{5} + 16 x + 32 x^{3}) (x^{2} {(x^{4} + 4)}^{3} + 2 {(x^{4} + 4)}^{3})$ by multiplying each term in the first expression by each term in the second expression.

$y' = 20 x^{5} (x^{2} {(x^{4} + 4)}^{3}) + 20 x^{5} (2 {(x^{4} + 4)}^{3}) + 16 x (x^{2} {(x^{4} + 4)}^{3}) + 16 x (2 {(x^{4} + 4)}^{3}) + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6

Simplify each term.

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Step 5.6.1

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

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Step 5.6.1.1

Move $x^{2}$ .

$y' = 20 (x^{2} x^{5}) {(x^{4} + 4)}^{3} + 20 x^{5} (2 {(x^{4} + 4)}^{3}) + 16 x (x^{2} {(x^{4} + 4)}^{3}) + 16 x (2 {(x^{4} + 4)}^{3}) + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.1.2

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

$y' = 20 x^{2 + 5} {(x^{4} + 4)}^{3} + 20 x^{5} (2 {(x^{4} + 4)}^{3}) + 16 x (x^{2} {(x^{4} + 4)}^{3}) + 16 x (2 {(x^{4} + 4)}^{3}) + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.1.3

Add $2$ and $5$ .

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 20 x^{5} (2 {(x^{4} + 4)}^{3}) + 16 x (x^{2} {(x^{4} + 4)}^{3}) + 16 x (2 {(x^{4} + 4)}^{3}) + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.2

Rewrite using the commutative property of multiplication.

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 20 \cdot 2 x^{5} {(x^{4} + 4)}^{3} + 16 x (x^{2} {(x^{4} + 4)}^{3}) + 16 x (2 {(x^{4} + 4)}^{3}) + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.3

Multiply $20$ by $2$ .

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 x (x^{2} {(x^{4} + 4)}^{3}) + 16 x (2 {(x^{4} + 4)}^{3}) + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.4

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

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Step 5.6.4.1

Move $x^{2}$ .

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 (x^{2} x) {(x^{4} + 4)}^{3} + 16 x (2 {(x^{4} + 4)}^{3}) + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.4.2

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

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Step 5.6.4.2.1

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

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 (x^{2} x^{1}) {(x^{4} + 4)}^{3} + 16 x (2 {(x^{4} + 4)}^{3}) + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.4.2.2

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

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 x^{2 + 1} {(x^{4} + 4)}^{3} + 16 x (2 {(x^{4} + 4)}^{3}) + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.4.3

Add $2$ and $1$ .

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 x^{3} {(x^{4} + 4)}^{3} + 16 x (2 {(x^{4} + 4)}^{3}) + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.5

Rewrite using the commutative property of multiplication.

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 x^{3} {(x^{4} + 4)}^{3} + 16 \cdot 2 x {(x^{4} + 4)}^{3} + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.6

Multiply $16$ by $2$ .

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 x^{3} {(x^{4} + 4)}^{3} + 32 x {(x^{4} + 4)}^{3} + 32 x^{3} (x^{2} {(x^{4} + 4)}^{3}) + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.7

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 5.6.7.1

Move $x^{2}$ .

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 x^{3} {(x^{4} + 4)}^{3} + 32 x {(x^{4} + 4)}^{3} + 32 (x^{2} x^{3}) {(x^{4} + 4)}^{3} + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.7.2

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

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 x^{3} {(x^{4} + 4)}^{3} + 32 x {(x^{4} + 4)}^{3} + 32 x^{2 + 3} {(x^{4} + 4)}^{3} + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.7.3

Add $2$ and $3$ .

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 x^{3} {(x^{4} + 4)}^{3} + 32 x {(x^{4} + 4)}^{3} + 32 x^{5} {(x^{4} + 4)}^{3} + 32 x^{3} (2 {(x^{4} + 4)}^{3})$

Step 5.6.8

Rewrite using the commutative property of multiplication.

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 x^{3} {(x^{4} + 4)}^{3} + 32 x {(x^{4} + 4)}^{3} + 32 x^{5} {(x^{4} + 4)}^{3} + 32 \cdot 2 x^{3} {(x^{4} + 4)}^{3}$

Step 5.6.9

Multiply $32$ by $2$ .

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 40 x^{5} {(x^{4} + 4)}^{3} + 16 x^{3} {(x^{4} + 4)}^{3} + 32 x {(x^{4} + 4)}^{3} + 32 x^{5} {(x^{4} + 4)}^{3} + 64 x^{3} {(x^{4} + 4)}^{3}$

Step 5.7

Add $40 x^{5} {(x^{4} + 4)}^{3}$ and $32 x^{5} {(x^{4} + 4)}^{3}$ .

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 72 x^{5} {(x^{4} + 4)}^{3} + 16 x^{3} {(x^{4} + 4)}^{3} + 32 x {(x^{4} + 4)}^{3} + 64 x^{3} {(x^{4} + 4)}^{3}$

Step 5.8

Add $16 x^{3} {(x^{4} + 4)}^{3}$ and $64 x^{3} {(x^{4} + 4)}^{3}$ .

$y' = 20 x^{7} {(x^{4} + 4)}^{3} + 72 x^{5} {(x^{4} + 4)}^{3} + 80 x^{3} {(x^{4} + 4)}^{3} + 32 x {(x^{4} + 4)}^{3}$