Calculus Examples

$y = {(3 x^{4} + 1)}^{4} (x^{3} + 4)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(3 x^{4} + 1)}^{4}$ and $g (x) = x^{3} + 4$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [4]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $3 x^{2}$ and $0$ .

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

Step 2.4.2

Move $3$ to the left of ${(3 x^{4} + 1)}^{4}$ .

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

Step 3

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 $g (x) = 3 x^{4} + 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $3 x^{4} + 1$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $3 x^{4} + 1$ .

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

Step 4

Differentiate.

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Step 4.1

Move $4$ to the left of $x^{3} + 4$ .

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

Step 4.2

By the Sum Rule, the derivative of $3 x^{4} + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [1]$ .

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

Step 4.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

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

Step 4.4

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

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

Step 4.5

Multiply $4$ by $3$ .

$3 {(3 x^{4} + 1)}^{4} x^{2} + 4 (x^{3} + 4) {(3 x^{4} + 1)}^{3} (12 x^{3} + \frac{d}{d x} [1])$

Step 4.6

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

$3 {(3 x^{4} + 1)}^{4} x^{2} + 4 (x^{3} + 4) {(3 x^{4} + 1)}^{3} (12 x^{3} + 0)$

Step 4.7

Simplify the expression.

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Step 4.7.1

Add $12 x^{3}$ and $0$ .

$3 {(3 x^{4} + 1)}^{4} x^{2} + 4 (x^{3} + 4) {(3 x^{4} + 1)}^{3} (12 x^{3})$

Step 4.7.2

Multiply $12$ by $4$ .

$3 {(3 x^{4} + 1)}^{4} x^{2} + 48 (x^{3} + 4) {(3 x^{4} + 1)}^{3} x^{3}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$3 {(3 x^{4} + 1)}^{4} x^{2} + (48 x^{3} + 48 \cdot 4) {(3 x^{4} + 1)}^{3} x^{3}$

Step 5.2

Multiply $48$ by $4$ .

$3 {(3 x^{4} + 1)}^{4} x^{2} + (48 x^{3} + 192) {(3 x^{4} + 1)}^{3} x^{3}$

Step 5.3

Factor ${(3 x^{4} + 1)}^{3} x^{2}$ out of $3 {(3 x^{4} + 1)}^{4} x^{2} + (48 x^{3} + 192) {(3 x^{4} + 1)}^{3} x^{3}$ .

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Step 5.3.1

Factor ${(3 x^{4} + 1)}^{3} x^{2}$ out of $3 {(3 x^{4} + 1)}^{4} x^{2}$ .

${(3 x^{4} + 1)}^{3} x^{2} (3 (3 x^{4} + 1)) + (48 x^{3} + 192) {(3 x^{4} + 1)}^{3} x^{3}$

Step 5.3.2

Factor ${(3 x^{4} + 1)}^{3} x^{2}$ out of $(48 x^{3} + 192) {(3 x^{4} + 1)}^{3} x^{3}$ .

${(3 x^{4} + 1)}^{3} x^{2} (3 (3 x^{4} + 1)) + {(3 x^{4} + 1)}^{3} x^{2} ((48 x^{3} + 192) x)$

Step 5.3.3

Factor ${(3 x^{4} + 1)}^{3} x^{2}$ out of ${(3 x^{4} + 1)}^{3} x^{2} (3 (3 x^{4} + 1)) + {(3 x^{4} + 1)}^{3} x^{2} ((48 x^{3} + 192) x)$ .

${(3 x^{4} + 1)}^{3} x^{2} (3 (3 x^{4} + 1) + (48 x^{3} + 192) x)$

Step 5.4

Reorder the factors of ${(3 x^{4} + 1)}^{3} x^{2} (3 (3 x^{4} + 1) + (48 x^{3} + 192) x)$ .

$x^{2} {(3 x^{4} + 1)}^{3} (3 (3 x^{4} + 1) + (48 x^{3} + 192) x)$