Calculus Examples

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

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

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

Step 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 $g (x) = 3 x + 1$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x + 1$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $3 x + 1$ .

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

Step 3

Differentiate.

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Step 3.1

Move $4$ to the left of $4 x - 1$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply $3$ by $1$ .

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

Step 3.6

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

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

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $3$ and $0$ .

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

Step 3.7.2

Multiply $3$ by $4$ .

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Multiply $4$ by $1$ .

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

Step 3.12

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

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

Step 3.13

Simplify the expression.

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Step 3.13.1

Add $4$ and $0$ .

$12 (4 x - 1) {(3 x + 1)}^{3} + {(3 x + 1)}^{4} \cdot 4$

Step 3.13.2

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

$12 (4 x - 1) {(3 x + 1)}^{3} + 4 \cdot {(3 x + 1)}^{4}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(12 (4 x) + 12 \cdot - 1) {(3 x + 1)}^{3} + 4 {(3 x + 1)}^{4}$

Step 4.2

Multiply $4$ by $12$ .

$(48 x + 12 \cdot - 1) {(3 x + 1)}^{3} + 4 {(3 x + 1)}^{4}$

Step 4.3

Multiply $12$ by $- 1$ .

$(48 x - 12) {(3 x + 1)}^{3} + 4 {(3 x + 1)}^{4}$

Step 4.4

Factor ${(3 x + 1)}^{3}$ out of $(48 x - 12) {(3 x + 1)}^{3} + 4 {(3 x + 1)}^{4}$ .

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Step 4.4.1

Factor ${(3 x + 1)}^{3}$ out of $(48 x - 12) {(3 x + 1)}^{3}$ .

${(3 x + 1)}^{3} (48 x - 12) + 4 {(3 x + 1)}^{4}$

Step 4.4.2

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

${(3 x + 1)}^{3} (48 x - 12) + {(3 x + 1)}^{3} (4 (3 x + 1))$

Step 4.4.3

Factor ${(3 x + 1)}^{3}$ out of ${(3 x + 1)}^{3} (48 x - 12) + {(3 x + 1)}^{3} (4 (3 x + 1))$ .

${(3 x + 1)}^{3} (48 x - 12 + 4 (3 x + 1))$