Calculus Examples

$f' (x) = 4 x^{3} + 15 x^{2} + 10 x + 2$ , $x = 5$

Step 1

Find the derivative.

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Step 1.1

By the Sum Rule, the derivative of $4 x^{3} + 15 x^{2} + 10 x + 2$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [2]$

Step 1.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [2]$

Step 1.2.2

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

$4 (3 x^{2}) + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [2]$

Step 1.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [2]$

Step 1.3

Evaluate $\frac{d}{d x} [15 x^{2}]$ .

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Step 1.3.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{2}$ with respect to $x$ is $15 \frac{d}{d x} [x^{2}]$ .

$12 x^{2} + 15 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [2]$

Step 1.3.2

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

$12 x^{2} + 15 (2 x) + \frac{d}{d x} [10 x] + \frac{d}{d x} [2]$

Step 1.3.3

Multiply $2$ by $15$ .

$12 x^{2} + 30 x + \frac{d}{d x} [10 x] + \frac{d}{d x} [2]$

Step 1.4

Evaluate $\frac{d}{d x} [10 x]$ .

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Step 1.4.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$12 x^{2} + 30 x + 10 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 1.4.2

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

$12 x^{2} + 30 x + 10 \cdot 1 + \frac{d}{d x} [2]$

Step 1.4.3

Multiply $10$ by $1$ .

$12 x^{2} + 30 x + 10 + \frac{d}{d x} [2]$

Step 1.5

Differentiate using the Constant Rule.

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Step 1.5.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$12 x^{2} + 30 x + 10 + 0$

Step 1.5.2

Add $12 x^{2} + 30 x + 10$ and $0$ .

$12 x^{2} + 30 x + 10$

Step 2

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

$f'' (5) = 12 {(5)}^{2} + 30 (5) + 10$

Step 3

Simplify each term.

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Step 3.1

Raise $5$ to the power of $2$ .

$f'' (5) = 12 \cdot 25 + 30 (5) + 10$

Step 3.2

Multiply $12$ by $25$ .

$f'' (5) = 300 + 30 (5) + 10$

Step 3.3

Multiply $30$ by $5$ .

$f'' (5) = 300 + 150 + 10$

Step 4

Simplify by adding numbers.

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Step 4.1

Add $300$ and $150$ .

$f'' (5) = 450 + 10$

Step 4.2

Add $450$ and $10$ .

$f'' (5) = 460$