Precalculus Examples

$f (x) = 7 x + 8$ ; $x = 5$

Step 1

Find the derivative.

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Step 1.1

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

$\frac{d}{d x} [7 x] + \frac{d}{d x} [8]$

Step 1.2

Evaluate $\frac{d}{d x} [7 x]$ .

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Step 1.2.1

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

$7 \frac{d}{d x} [x] + \frac{d}{d x} [8]$

Step 1.2.2

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

$7 \cdot 1 + \frac{d}{d x} [8]$

Step 1.2.3

Multiply $7$ by $1$ .

$7 + \frac{d}{d x} [8]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

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

$7 + 0$

Step 1.3.2

Add $7$ and $0$ .

$7$

Step 2

Replace the variable $x$ with $5$ in the expression.

$f' (5) = 7$