Calculus Examples

$x (7 + 2 x)$

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 $f (x) = x$ and $g (x) = 7 + 2 x$ .

$x \frac{d}{d x} [7 + 2 x] + (7 + 2 x) \frac{d}{d x} [x]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $7 + 2 x$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [2 x]$ .

$x (\frac{d}{d x} [7] + \frac{d}{d x} [2 x]) + (7 + 2 x) \frac{d}{d x} [x]$

Step 2.2

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

$x (0 + \frac{d}{d x} [2 x]) + (7 + 2 x) \frac{d}{d x} [x]$

Step 2.3

Add $0$ and $\frac{d}{d x} [2 x]$ .

$x \frac{d}{d x} [2 x] + (7 + 2 x) \frac{d}{d x} [x]$

Step 2.4

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

$x (2 \frac{d}{d x} [x]) + (7 + 2 x) \frac{d}{d x} [x]$

Step 2.5

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

$x (2 \cdot 1) + (7 + 2 x) \frac{d}{d x} [x]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $2$ by $1$ .

$x \cdot 2 + (7 + 2 x) \frac{d}{d x} [x]$

Step 2.6.2

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

$2 \cdot x + (7 + 2 x) \frac{d}{d x} [x]$

Step 2.7

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

$2 x + (7 + 2 x) \cdot 1$

Step 2.8

Simplify by adding terms.

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Step 2.8.1

Multiply $7 + 2 x$ by $1$ .

$2 x + 7 + 2 x$

Step 2.8.2

Add $2 x$ and $2 x$ .

$4 x + 7$