Calculus Examples

$f (x) = 3 x^{3} - 5 x^{2} + 5 x + 7$ , $f' (6)$

Step 1

Find the derivative of $f (x) = 3 x^{3} - 5 x^{2} + 5 x + 7$ .

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Step 1.1

By the Sum Rule, the derivative of $3 x^{3} - 5 x^{2} + 5 x + 7$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [7]$

Step 1.2

Evaluate $\frac{d}{d x} [3 x^{3}]$ .

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Step 1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x^{3}]$ .

$3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [7]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 (3 x^{2}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [7]$

Step 1.2.3

Multiply $3$ by $3$ .

$9 x^{2} + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [7]$

Step 1.3

Evaluate $\frac{d}{d x} [- 5 x^{2}]$ .

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Step 1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

$9 x^{2} - 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [7]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$9 x^{2} - 5 (2 x) + \frac{d}{d x} [5 x] + \frac{d}{d x} [7]$

Step 1.3.3

Multiply $2$ by $- 5$ .

$9 x^{2} - 10 x + \frac{d}{d x} [5 x] + \frac{d}{d x} [7]$

Step 1.4

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 1.4.1

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

$9 x^{2} - 10 x + 5 \frac{d}{d x} [x] + \frac{d}{d x} [7]$

Step 1.4.2

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

$9 x^{2} - 10 x + 5 \cdot 1 + \frac{d}{d x} [7]$

Step 1.4.3

Multiply $5$ by $1$ .

$9 x^{2} - 10 x + 5 + \frac{d}{d x} [7]$

Step 1.5

Differentiate using the Constant Rule.

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Step 1.5.1

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

$9 x^{2} - 10 x + 5 + 0$

Step 1.5.2

Add $9 x^{2} - 10 x + 5$ and $0$ .

$9 x^{2} - 10 x + 5$

Step 2

Replace the variable $x$ with $6$ in the expression.

$9 {(6)}^{2} - 10 \cdot 6 + 5$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Raise $6$ to the power of $2$ .

$9 \cdot 36 - 10 \cdot 6 + 5$

Step 3.1.2

Multiply $9$ by $36$ .

$324 - 10 \cdot 6 + 5$

Step 3.1.3

Multiply $- 10$ by $6$ .

$324 - 60 + 5$

Step 3.2

Simplify by adding and subtracting.

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Step 3.2.1

Subtract $60$ from $324$ .

$264 + 5$

Step 3.2.2

Add $264$ and $5$ .

$269$