Calculus Examples

$H (x) = 4 f (x) + 2 g (x) + 9$

Step 1

By the Sum Rule, the derivative of $4 f (x) + 2 g (x) + 9$ with respect to $x$ is $\frac{d}{d x} [4 f (x)] + \frac{d}{d x} [2 g (x)] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [4 f (x)] + \frac{d}{d x} [2 g (x)] + \frac{d}{d x} [9]$

Step 2

Evaluate $\frac{d}{d x} [4 f (x)]$ .

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Step 2.1

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

$4 f \frac{d}{d x} [x] + \frac{d}{d x} [2 g (x)] + \frac{d}{d x} [9]$

Step 2.2

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

$4 f \cdot 1 + \frac{d}{d x} [2 g (x)] + \frac{d}{d x} [9]$

Step 2.3

Multiply $4$ by $1$ .

$4 f + \frac{d}{d x} [2 g (x)] + \frac{d}{d x} [9]$

Step 3

Evaluate $\frac{d}{d x} [2 g (x)]$ .

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Step 3.1

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

$4 f + 2 g \frac{d}{d x} [x] + \frac{d}{d x} [9]$

Step 3.2

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

$4 f + 2 g \cdot 1 + \frac{d}{d x} [9]$

Step 3.3

Multiply $2$ by $1$ .

$4 f + 2 g + \frac{d}{d x} [9]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$4 f + 2 g + 0$

Step 4.2

Add $4 f + 2 g$ and $0$ .

$4 f + 2 g$