Calculus Examples

$(x + 6) (4 x - 3)$

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 $f (x) = x + 6$ and $g (x) = 4 x - 3$ .

$(x + 6) \frac{d}{d x} [4 x - 3] + (4 x - 3) \frac{d}{d x} [x + 6]$

Step 2

Differentiate.

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Step 2.1

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

$(x + 6) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]) + (4 x - 3) \frac{d}{d x} [x + 6]$

Step 2.2

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

$(x + 6) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) + (4 x - 3) \frac{d}{d x} [x + 6]$

Step 2.3

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

$(x + 6) (4 \cdot 1 + \frac{d}{d x} [- 3]) + (4 x - 3) \frac{d}{d x} [x + 6]$

Step 2.4

Multiply $4$ by $1$ .

$(x + 6) (4 + \frac{d}{d x} [- 3]) + (4 x - 3) \frac{d}{d x} [x + 6]$

Step 2.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$(x + 6) (4 + 0) + (4 x - 3) \frac{d}{d x} [x + 6]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $4$ and $0$ .

$(x + 6) \cdot 4 + (4 x - 3) \frac{d}{d x} [x + 6]$

Step 2.6.2

Move $4$ to the left of $x + 6$ .

$4 \cdot (x + 6) + (4 x - 3) \frac{d}{d x} [x + 6]$

Step 2.7

By the Sum Rule, the derivative of $x + 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [6]$ .

$4 (x + 6) + (4 x - 3) (\frac{d}{d x} [x] + \frac{d}{d x} [6])$

Step 2.8

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

$4 (x + 6) + (4 x - 3) (1 + \frac{d}{d x} [6])$

Step 2.9

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$4 (x + 6) + (4 x - 3) (1 + 0)$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $1$ and $0$ .

$4 (x + 6) + (4 x - 3) \cdot 1$

Step 2.10.2

Multiply $4 x - 3$ by $1$ .

$4 (x + 6) + 4 x - 3$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$4 x + 4 \cdot 6 + 4 x - 3$

Step 3.2

Combine terms.

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Step 3.2.1

Multiply $4$ by $6$ .

$4 x + 24 + 4 x - 3$

Step 3.2.2

Add $4 x$ and $4 x$ .

$8 x + 24 - 3$

Step 3.2.3

Subtract $3$ from $24$ .

$8 x + 21$