Calculus Examples

$y = (5 x - 1) (4 + 3 x)$

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 $f (x) = 5 x - 1$ and $g (x) = 4 + 3 x$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Add $0$ and $\frac{d}{d x} [3 x]$ .

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

Step 2.4

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

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

Step 2.5

Move $3$ to the left of $5 x - 1$ .

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

Step 2.6

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

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

Step 2.7

Multiply $3$ by $1$ .

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

Step 2.8

By the Sum Rule, the derivative of $5 x - 1$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 1]$ .

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

Step 2.9

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

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

Step 2.10

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

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

Step 2.11

Multiply $5$ by $1$ .

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

Step 2.12

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

$3 (5 x - 1) + (4 + 3 x) (5 + 0)$

Step 2.13

Simplify the expression.

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Step 2.13.1

Add $5$ and $0$ .

$3 (5 x - 1) + (4 + 3 x) \cdot 5$

Step 2.13.2

Move $5$ to the left of $4 + 3 x$ .

$3 (5 x - 1) + 5 \cdot (4 + 3 x)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$3 (5 x) + 3 \cdot - 1 + 5 (4 + 3 x)$

Step 3.2

Apply the distributive property.

$3 (5 x) + 3 \cdot - 1 + 5 \cdot 4 + 5 (3 x)$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $5$ by $3$ .

$15 x + 3 \cdot - 1 + 5 \cdot 4 + 5 (3 x)$

Step 3.3.2

Multiply $3$ by $- 1$ .

$15 x - 3 + 5 \cdot 4 + 5 (3 x)$

Step 3.3.3

Multiply $5$ by $4$ .

$15 x - 3 + 20 + 5 (3 x)$

Step 3.3.4

Multiply $3$ by $5$ .

$15 x - 3 + 20 + 15 x$

Step 3.3.5

Add $- 3$ and $20$ .

$15 x + 17 + 15 x$

Step 3.3.6

Add $15 x$ and $15 x$ .

$30 x + 17$