Calculus Examples

$y = (2 x - 3) (x + 5)$

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 $f (x) = 2 x - 3$ and $g (x) = x + 5$ .

$(2 x - 3) \frac{d}{d x} [x + 5] + (x + 5) \frac{d}{d x} [2 x - 3]$

Step 2

Differentiate.

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Step 2.1

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

$(2 x - 3) (\frac{d}{d x} [x] + \frac{d}{d x} [5]) + (x + 5) \frac{d}{d x} [2 x - 3]$

Step 2.2

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

$(2 x - 3) (1 + \frac{d}{d x} [5]) + (x + 5) \frac{d}{d x} [2 x - 3]$

Step 2.3

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

$(2 x - 3) (1 + 0) + (x + 5) \frac{d}{d x} [2 x - 3]$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

$(2 x - 3) \cdot 1 + (x + 5) \frac{d}{d x} [2 x - 3]$

Step 2.4.2

Multiply $2 x - 3$ by $1$ .

$2 x - 3 + (x + 5) \frac{d}{d x} [2 x - 3]$

Step 2.5

By the Sum Rule, the derivative of $2 x - 3$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]$ .

$2 x - 3 + (x + 5) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3])$

Step 2.6

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 x - 3 + (x + 5) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 2.7

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

$2 x - 3 + (x + 5) (2 \cdot 1 + \frac{d}{d x} [- 3])$

Step 2.8

Multiply $2$ by $1$ .

$2 x - 3 + (x + 5) (2 + \frac{d}{d x} [- 3])$

Step 2.9

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

$2 x - 3 + (x + 5) (2 + 0)$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2$ and $0$ .

$2 x - 3 + (x + 5) \cdot 2$

Step 2.10.2

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

$2 x - 3 + 2 \cdot (x + 5)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$2 x - 3 + 2 x + 2 \cdot 5$

Step 3.2

Combine terms.

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Step 3.2.1

Multiply $2$ by $5$ .

$2 x - 3 + 2 x + 10$

Step 3.2.2

Add $2 x$ and $2 x$ .

$4 x - 3 + 10$

Step 3.2.3

Add $- 3$ and $10$ .

$4 x + 7$