Calculus Examples

$(2 + 5 x) (5 - 4 x)$

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 $f (x) = 2 + 5 x$ and $g (x) = 5 - 4 x$ .

$(2 + 5 x) \frac{d}{d x} [5 - 4 x] + (5 - 4 x) \frac{d}{d x} [2 + 5 x]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $5 - 4 x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 4 x]$ .

$(2 + 5 x) (\frac{d}{d x} [5] + \frac{d}{d x} [- 4 x]) + (5 - 4 x) \frac{d}{d x} [2 + 5 x]$

Step 2.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$(2 + 5 x) (0 + \frac{d}{d x} [- 4 x]) + (5 - 4 x) \frac{d}{d x} [2 + 5 x]$

Step 2.3

Add $0$ and $\frac{d}{d x} [- 4 x]$ .

$(2 + 5 x) \frac{d}{d x} [- 4 x] + (5 - 4 x) \frac{d}{d x} [2 + 5 x]$

Step 2.4

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

$(2 + 5 x) (- 4 \frac{d}{d x} [x]) + (5 - 4 x) \frac{d}{d x} [2 + 5 x]$

Step 2.5

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

$(2 + 5 x) (- 4 \cdot 1) + (5 - 4 x) \frac{d}{d x} [2 + 5 x]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $- 4$ by $1$ .

$(2 + 5 x) \cdot - 4 + (5 - 4 x) \frac{d}{d x} [2 + 5 x]$

Step 2.6.2

Move $- 4$ to the left of $2 + 5 x$ .

$- 4 \cdot (2 + 5 x) + (5 - 4 x) \frac{d}{d x} [2 + 5 x]$

Step 2.7

By the Sum Rule, the derivative of $2 + 5 x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [5 x]$ .

$- 4 (2 + 5 x) + (5 - 4 x) (\frac{d}{d x} [2] + \frac{d}{d x} [5 x])$

Step 2.8

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$- 4 (2 + 5 x) + (5 - 4 x) (0 + \frac{d}{d x} [5 x])$

Step 2.9

Add $0$ and $\frac{d}{d x} [5 x]$ .

$- 4 (2 + 5 x) + (5 - 4 x) \frac{d}{d x} [5 x]$

Step 2.10

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

$- 4 (2 + 5 x) + (5 - 4 x) (5 \frac{d}{d x} [x])$

Step 2.11

Move $5$ to the left of $5 - 4 x$ .

$- 4 (2 + 5 x) + 5 \cdot (5 - 4 x) \frac{d}{d x} [x]$

Step 2.12

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

$- 4 (2 + 5 x) + 5 (5 - 4 x) \cdot 1$

Step 2.13

Multiply $5$ by $1$ .

$- 4 (2 + 5 x) + 5 (5 - 4 x)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$- 4 \cdot 2 - 4 (5 x) + 5 (5 - 4 x)$

Step 3.2

Apply the distributive property.

$- 4 \cdot 2 - 4 (5 x) + 5 \cdot 5 + 5 (- 4 x)$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $- 4$ by $2$ .

$- 8 - 4 (5 x) + 5 \cdot 5 + 5 (- 4 x)$

Step 3.3.2

Multiply $5$ by $- 4$ .

$- 8 - 20 x + 5 \cdot 5 + 5 (- 4 x)$

Step 3.3.3

Multiply $5$ by $5$ .

$- 8 - 20 x + 25 + 5 (- 4 x)$

Step 3.3.4

Multiply $- 4$ by $5$ .

$- 8 - 20 x + 25 - 20 x$

Step 3.3.5

Add $- 8$ and $25$ .

$- 20 x + 17 - 20 x$

Step 3.3.6

Subtract $20 x$ from $- 20 x$ .

$- 40 x + 17$