Calculus Examples

$f (x) = (- 5 x^{2} - 5 x - 2) (- 4 x - 5)$ , $x = 0$

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 $f (x) = - 5 x^{2} - 5 x - 2$ and $g (x) = - 4 x - 5$ .

$(- 5 x^{2} - 5 x - 2) \frac{d}{d x} [- 4 x - 5] + (- 4 x - 5) \frac{d}{d x} [- 5 x^{2} - 5 x - 2]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $- 4 x - 5$ with respect to $x$ is $\frac{d}{d x} [- 4 x] + \frac{d}{d x} [- 5]$ .

$(- 5 x^{2} - 5 x - 2) (\frac{d}{d x} [- 4 x] + \frac{d}{d x} [- 5]) + (- 4 x - 5) \frac{d}{d x} [- 5 x^{2} - 5 x - 2]$

Step 2.2

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

$(- 5 x^{2} - 5 x - 2) (- 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]) + (- 4 x - 5) \frac{d}{d x} [- 5 x^{2} - 5 x - 2]$

Step 2.3

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

$(- 5 x^{2} - 5 x - 2) (- 4 \cdot 1 + \frac{d}{d x} [- 5]) + (- 4 x - 5) \frac{d}{d x} [- 5 x^{2} - 5 x - 2]$

Step 2.4

Multiply $- 4$ by $1$ .

$(- 5 x^{2} - 5 x - 2) (- 4 + \frac{d}{d x} [- 5]) + (- 4 x - 5) \frac{d}{d x} [- 5 x^{2} - 5 x - 2]$

Step 2.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$(- 5 x^{2} - 5 x - 2) (- 4 + 0) + (- 4 x - 5) \frac{d}{d x} [- 5 x^{2} - 5 x - 2]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $- 4$ and $0$ .

$(- 5 x^{2} - 5 x - 2) \cdot - 4 + (- 4 x - 5) \frac{d}{d x} [- 5 x^{2} - 5 x - 2]$

Step 2.6.2

Move $- 4$ to the left of $- 5 x^{2} - 5 x - 2$ .

$- 4 \cdot (- 5 x^{2} - 5 x - 2) + (- 4 x - 5) \frac{d}{d x} [- 5 x^{2} - 5 x - 2]$

Step 2.7

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

$- 4 (- 5 x^{2} - 5 x - 2) + (- 4 x - 5) (\frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 2])$

Step 2.8

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

$- 4 (- 5 x^{2} - 5 x - 2) + (- 4 x - 5) (- 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 2])$

Step 2.9

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

$- 4 (- 5 x^{2} - 5 x - 2) + (- 4 x - 5) (- 5 (2 x) + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 2])$

Step 2.10

Multiply $2$ by $- 5$ .

$- 4 (- 5 x^{2} - 5 x - 2) + (- 4 x - 5) (- 10 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 2])$

Step 2.11

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

$- 4 (- 5 x^{2} - 5 x - 2) + (- 4 x - 5) (- 10 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 2])$

Step 2.12

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

$- 4 (- 5 x^{2} - 5 x - 2) + (- 4 x - 5) (- 10 x - 5 \cdot 1 + \frac{d}{d x} [- 2])$

Step 2.13

Multiply $- 5$ by $1$ .

$- 4 (- 5 x^{2} - 5 x - 2) + (- 4 x - 5) (- 10 x - 5 + \frac{d}{d x} [- 2])$

Step 2.14

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

$- 4 (- 5 x^{2} - 5 x - 2) + (- 4 x - 5) (- 10 x - 5 + 0)$

Step 2.15

Add $- 10 x - 5$ and $0$ .

$- 4 (- 5 x^{2} - 5 x - 2) + (- 4 x - 5) (- 10 x - 5)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$- 4 (- 5 x^{2}) - 4 (- 5 x) - 4 \cdot - 2 + (- 4 x - 5) (- 10 x - 5)$

Step 3.2

Apply the distributive property.

$- 4 (- 5 x^{2}) - 4 (- 5 x) - 4 \cdot - 2 + (- 4 x - 5) (- 10 x) + (- 4 x - 5) \cdot - 5$

Step 3.3

Apply the distributive property.

$- 4 (- 5 x^{2}) - 4 (- 5 x) - 4 \cdot - 2 - 4 x (- 10 x) - 5 (- 10 x) + (- 4 x - 5) \cdot - 5$

Step 3.4

Apply the distributive property.

$- 4 (- 5 x^{2}) - 4 (- 5 x) - 4 \cdot - 2 - 4 x (- 10 x) - 5 (- 10 x) - 4 x \cdot - 5 - 5 \cdot - 5$

Step 3.5

Combine terms.

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Step 3.5.1

Multiply $- 5$ by $- 4$ .

$20 x^{2} - 4 (- 5 x) - 4 \cdot - 2 - 4 x (- 10 x) - 5 (- 10 x) - 4 x \cdot - 5 - 5 \cdot - 5$

Step 3.5.2

Multiply $- 5$ by $- 4$ .

$20 x^{2} + 20 x - 4 \cdot - 2 - 4 x (- 10 x) - 5 (- 10 x) - 4 x \cdot - 5 - 5 \cdot - 5$

Step 3.5.3

Multiply $- 4$ by $- 2$ .

$20 x^{2} + 20 x + 8 - 4 x (- 10 x) - 5 (- 10 x) - 4 x \cdot - 5 - 5 \cdot - 5$

Step 3.5.4

Multiply $- 10$ by $- 4$ .

$20 x^{2} + 20 x + 8 + 40 x \cdot x - 5 (- 10 x) - 4 x \cdot - 5 - 5 \cdot - 5$

Step 3.5.5

Raise $x$ to the power of $1$ .

$20 x^{2} + 20 x + 8 + 40 (x^{1} x) - 5 (- 10 x) - 4 x \cdot - 5 - 5 \cdot - 5$

Step 3.5.6

Raise $x$ to the power of $1$ .

$20 x^{2} + 20 x + 8 + 40 (x^{1} x^{1}) - 5 (- 10 x) - 4 x \cdot - 5 - 5 \cdot - 5$

Step 3.5.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$20 x^{2} + 20 x + 8 + 40 x^{1 + 1} - 5 (- 10 x) - 4 x \cdot - 5 - 5 \cdot - 5$

Step 3.5.8

Add $1$ and $1$ .

$20 x^{2} + 20 x + 8 + 40 x^{2} - 5 (- 10 x) - 4 x \cdot - 5 - 5 \cdot - 5$

Step 3.5.9

Multiply $- 10$ by $- 5$ .

$20 x^{2} + 20 x + 8 + 40 x^{2} + 50 x - 4 x \cdot - 5 - 5 \cdot - 5$

Step 3.5.10

Multiply $- 5$ by $- 4$ .

$20 x^{2} + 20 x + 8 + 40 x^{2} + 50 x + 20 x - 5 \cdot - 5$

Step 3.5.11

Multiply $- 5$ by $- 5$ .

$20 x^{2} + 20 x + 8 + 40 x^{2} + 50 x + 20 x + 25$

Step 3.5.12

Add $50 x$ and $20 x$ .

$20 x^{2} + 20 x + 8 + 40 x^{2} + 70 x + 25$

Step 3.5.13

Add $20 x^{2}$ and $40 x^{2}$ .

$60 x^{2} + 20 x + 8 + 70 x + 25$

Step 3.5.14

Add $20 x$ and $70 x$ .

$60 x^{2} + 90 x + 8 + 25$

Step 3.5.15

Add $8$ and $25$ .

$60 x^{2} + 90 x + 33$

Step 4

Evaluate the derivative at $x = 0$ .

$60 {(0)}^{2} + 90 (0) + 33$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Raising $0$ to any positive power yields $0$ .

$60 \cdot 0 + 90 (0) + 33$

Step 5.1.2

Multiply $60$ by $0$ .

$0 + 90 (0) + 33$

Step 5.1.3

Multiply $90$ by $0$ .

$0 + 0 + 33$

Step 5.2

Simplify by adding numbers.

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Step 5.2.1

Add $0$ and $0$ .

$0 + 33$

Step 5.2.2

Add $0$ and $33$ .

$33$