Calculus Examples

$f (x) = (6 x + 7) (7 x - 8)$

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 $f (x) = 6 x + 7$ and $g (x) = 7 x - 8$ .

$(6 x + 7) \frac{d}{d x} [7 x - 8] + (7 x - 8) \frac{d}{d x} [6 x + 7]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $7 x - 8$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [- 8]$ .

$(6 x + 7) (\frac{d}{d x} [7 x] + \frac{d}{d x} [- 8]) + (7 x - 8) \frac{d}{d x} [6 x + 7]$

Step 2.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$(6 x + 7) (7 \frac{d}{d x} [x] + \frac{d}{d x} [- 8]) + (7 x - 8) \frac{d}{d x} [6 x + 7]$

Step 2.3

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

$(6 x + 7) (7 \cdot 1 + \frac{d}{d x} [- 8]) + (7 x - 8) \frac{d}{d x} [6 x + 7]$

Step 2.4

Multiply $7$ by $1$ .

$(6 x + 7) (7 + \frac{d}{d x} [- 8]) + (7 x - 8) \frac{d}{d x} [6 x + 7]$

Step 2.5

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$(6 x + 7) (7 + 0) + (7 x - 8) \frac{d}{d x} [6 x + 7]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $7$ and $0$ .

$(6 x + 7) \cdot 7 + (7 x - 8) \frac{d}{d x} [6 x + 7]$

Step 2.6.2

Move $7$ to the left of $6 x + 7$ .

$7 \cdot (6 x + 7) + (7 x - 8) \frac{d}{d x} [6 x + 7]$

Step 2.7

By the Sum Rule, the derivative of $6 x + 7$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [7]$ .

$7 (6 x + 7) + (7 x - 8) (\frac{d}{d x} [6 x] + \frac{d}{d x} [7])$

Step 2.8

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$7 (6 x + 7) + (7 x - 8) (6 \frac{d}{d x} [x] + \frac{d}{d x} [7])$

Step 2.9

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

$7 (6 x + 7) + (7 x - 8) (6 \cdot 1 + \frac{d}{d x} [7])$

Step 2.10

Multiply $6$ by $1$ .

$7 (6 x + 7) + (7 x - 8) (6 + \frac{d}{d x} [7])$

Step 2.11

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$7 (6 x + 7) + (7 x - 8) (6 + 0)$

Step 2.12

Simplify the expression.

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Step 2.12.1

Add $6$ and $0$ .

$7 (6 x + 7) + (7 x - 8) \cdot 6$

Step 2.12.2

Move $6$ to the left of $7 x - 8$ .

$7 (6 x + 7) + 6 \cdot (7 x - 8)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$7 (6 x) + 7 \cdot 7 + 6 (7 x - 8)$

Step 3.2

Apply the distributive property.

$7 (6 x) + 7 \cdot 7 + 6 (7 x) + 6 \cdot - 8$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $6$ by $7$ .

$42 x + 7 \cdot 7 + 6 (7 x) + 6 \cdot - 8$

Step 3.3.2

Multiply $7$ by $7$ .

$42 x + 49 + 6 (7 x) + 6 \cdot - 8$

Step 3.3.3

Multiply $7$ by $6$ .

$42 x + 49 + 42 x + 6 \cdot - 8$

Step 3.3.4

Multiply $6$ by $- 8$ .

$42 x + 49 + 42 x - 48$

Step 3.3.5

Add $42 x$ and $42 x$ .

$84 x + 49 - 48$

Step 3.3.6

Subtract $48$ from $49$ .

$84 x + 1$