Calculus Examples

$y = (3 x^{2} - 5 x + 1) (2 - 3 x - x^{2})$ , $x = 4$

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 $f (x) = 3 x^{2} - 5 x + 1$ and $g (x) = 2 - 3 x - x^{2}$ .

$(3 x^{2} - 5 x + 1) \frac{d}{d x} [2 - 3 x - x^{2}] + (2 - 3 x - x^{2}) \frac{d}{d x} [3 x^{2} - 5 x + 1]$

Step 2

Differentiate.

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Step 2.1

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

$(3 x^{2} - 5 x + 1) (\frac{d}{d x} [2] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- x^{2}]) + (2 - 3 x - x^{2}) \frac{d}{d x} [3 x^{2} - 5 x + 1]$

Step 2.2

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

$(3 x^{2} - 5 x + 1) (0 + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- x^{2}]) + (2 - 3 x - x^{2}) \frac{d}{d x} [3 x^{2} - 5 x + 1]$

Step 2.3

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

$(3 x^{2} - 5 x + 1) (\frac{d}{d x} [- 3 x] + \frac{d}{d x} [- x^{2}]) + (2 - 3 x - x^{2}) \frac{d}{d x} [3 x^{2} - 5 x + 1]$

Step 2.4

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

$(3 x^{2} - 5 x + 1) (- 3 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}]) + (2 - 3 x - x^{2}) \frac{d}{d x} [3 x^{2} - 5 x + 1]$

Step 2.5

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

$(3 x^{2} - 5 x + 1) (- 3 \cdot 1 + \frac{d}{d x} [- x^{2}]) + (2 - 3 x - x^{2}) \frac{d}{d x} [3 x^{2} - 5 x + 1]$

Step 2.6

Multiply $- 3$ by $1$ .

$(3 x^{2} - 5 x + 1) (- 3 + \frac{d}{d x} [- x^{2}]) + (2 - 3 x - x^{2}) \frac{d}{d x} [3 x^{2} - 5 x + 1]$

Step 2.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$(3 x^{2} - 5 x + 1) (- 3 - \frac{d}{d x} [x^{2}]) + (2 - 3 x - x^{2}) \frac{d}{d x} [3 x^{2} - 5 x + 1]$

Step 2.8

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

$(3 x^{2} - 5 x + 1) (- 3 - (2 x)) + (2 - 3 x - x^{2}) \frac{d}{d x} [3 x^{2} - 5 x + 1]$

Step 2.9

Multiply $2$ by $- 1$ .

$(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) \frac{d}{d x} [3 x^{2} - 5 x + 1]$

Step 2.10

By the Sum Rule, the derivative of $3 x^{2} - 5 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [1]$ .

$(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [1])$

Step 2.11

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [1])$

Step 2.12

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

$(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (3 (2 x) + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [1])$

Step 2.13

Multiply $2$ by $3$ .

$(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (6 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [1])$

Step 2.14

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

$(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (6 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 2.15

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

$(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (6 x - 5 \cdot 1 + \frac{d}{d x} [1])$

Step 2.16

Multiply $- 5$ by $1$ .

$(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (6 x - 5 + \frac{d}{d x} [1])$

Step 2.17

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (6 x - 5 + 0)$

Step 2.18

Add $6 x - 5$ and $0$ .

$(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3

Simplify.

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Step 3.1

Factor $1$ out of $(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (6 x - 5)$ .

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Step 3.1.1

Factor $1$ out of $(3 x^{2} - 5 x + 1) (- 3 - 2 x)$ .

$1 ((3 x^{2} - 5 x + 1) (- 3 - 2 x)) + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.1.2

Factor $1$ out of $(2 - 3 x - x^{2}) (6 x - 5)$ .

$1 ((3 x^{2} - 5 x + 1) (- 3 - 2 x)) + 1 ((2 - 3 x - x^{2}) (6 x - 5))$

Step 3.1.3

Factor $1$ out of $1 ((3 x^{2} - 5 x + 1) (- 3 - 2 x)) + 1 ((2 - 3 x - x^{2}) (6 x - 5))$ .

$1 ((3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (6 x - 5))$

Step 3.2

Multiply $(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (6 x - 5)$ by $1$ .

$(3 x^{2} - 5 x + 1) (- 3 - 2 x) + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.3

Reorder terms.

$(- 3 - 2 x) (3 x^{2} - 5 x + 1) + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4

Simplify each term.

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Step 3.4.1

Expand $(- 3 - 2 x) (3 x^{2} - 5 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$- 3 (3 x^{2}) - 3 (- 5 x) - 3 \cdot 1 - 2 x (3 x^{2}) - 2 x (- 5 x) - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2

Simplify each term.

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Step 3.4.2.1

Multiply $3$ by $- 3$ .

$- 9 x^{2} - 3 (- 5 x) - 3 \cdot 1 - 2 x (3 x^{2}) - 2 x (- 5 x) - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.2

Multiply $- 5$ by $- 3$ .

$- 9 x^{2} + 15 x - 3 \cdot 1 - 2 x (3 x^{2}) - 2 x (- 5 x) - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.3

Multiply $- 3$ by $1$ .

$- 9 x^{2} + 15 x - 3 - 2 x (3 x^{2}) - 2 x (- 5 x) - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.4

Rewrite using the commutative property of multiplication.

$- 9 x^{2} + 15 x - 3 - 2 \cdot 3 x \cdot x^{2} - 2 x (- 5 x) - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.5

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.4.2.5.1

Move $x^{2}$ .

$- 9 x^{2} + 15 x - 3 - 2 \cdot 3 (x^{2} x) - 2 x (- 5 x) - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.5.2

Multiply $x^{2}$ by $x$ .

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Step 3.4.2.5.2.1

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

$- 9 x^{2} + 15 x - 3 - 2 \cdot 3 (x^{2} x^{1}) - 2 x (- 5 x) - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.5.2.2

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

$- 9 x^{2} + 15 x - 3 - 2 \cdot 3 x^{2 + 1} - 2 x (- 5 x) - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.5.3

Add $2$ and $1$ .

$- 9 x^{2} + 15 x - 3 - 2 \cdot 3 x^{3} - 2 x (- 5 x) - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.6

Multiply $- 2$ by $3$ .

$- 9 x^{2} + 15 x - 3 - 6 x^{3} - 2 x (- 5 x) - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.7

Rewrite using the commutative property of multiplication.

$- 9 x^{2} + 15 x - 3 - 6 x^{3} - 2 \cdot - 5 x \cdot x - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.8

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.2.8.1

Move $x$ .

$- 9 x^{2} + 15 x - 3 - 6 x^{3} - 2 \cdot - 5 (x \cdot x) - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.8.2

Multiply $x$ by $x$ .

$- 9 x^{2} + 15 x - 3 - 6 x^{3} - 2 \cdot - 5 x^{2} - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.9

Multiply $- 2$ by $- 5$ .

$- 9 x^{2} + 15 x - 3 - 6 x^{3} + 10 x^{2} - 2 x \cdot 1 + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.2.10

Multiply $- 2$ by $1$ .

$- 9 x^{2} + 15 x - 3 - 6 x^{3} + 10 x^{2} - 2 x + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.3

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

$x^{2} + 15 x - 3 - 6 x^{3} - 2 x + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.4

Subtract $2 x$ from $15 x$ .

$x^{2} + 13 x - 3 - 6 x^{3} + (2 - 3 x - x^{2}) (6 x - 5)$

Step 3.4.5

Expand $(2 - 3 x - x^{2}) (6 x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} + 13 x - 3 - 6 x^{3} + 2 (6 x) + 2 \cdot - 5 - 3 x (6 x) - 3 x \cdot - 5 - x^{2} (6 x) - x^{2} \cdot - 5$

Step 3.4.6

Simplify each term.

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Step 3.4.6.1

Multiply $6$ by $2$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x + 2 \cdot - 5 - 3 x (6 x) - 3 x \cdot - 5 - x^{2} (6 x) - x^{2} \cdot - 5$

Step 3.4.6.2

Multiply $2$ by $- 5$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 3 x (6 x) - 3 x \cdot - 5 - x^{2} (6 x) - x^{2} \cdot - 5$

Step 3.4.6.3

Rewrite using the commutative property of multiplication.

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 3 \cdot 6 x \cdot x - 3 x \cdot - 5 - x^{2} (6 x) - x^{2} \cdot - 5$

Step 3.4.6.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.6.4.1

Move $x$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 3 \cdot 6 (x \cdot x) - 3 x \cdot - 5 - x^{2} (6 x) - x^{2} \cdot - 5$

Step 3.4.6.4.2

Multiply $x$ by $x$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 3 \cdot 6 x^{2} - 3 x \cdot - 5 - x^{2} (6 x) - x^{2} \cdot - 5$

Step 3.4.6.5

Multiply $- 3$ by $6$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 18 x^{2} - 3 x \cdot - 5 - x^{2} (6 x) - x^{2} \cdot - 5$

Step 3.4.6.6

Multiply $- 5$ by $- 3$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 18 x^{2} + 15 x - x^{2} (6 x) - x^{2} \cdot - 5$

Step 3.4.6.7

Rewrite using the commutative property of multiplication.

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 18 x^{2} + 15 x - 1 \cdot 6 x^{2} x - x^{2} \cdot - 5$

Step 3.4.6.8

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.4.6.8.1

Move $x$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 18 x^{2} + 15 x - 1 \cdot 6 (x \cdot x^{2}) - x^{2} \cdot - 5$

Step 3.4.6.8.2

Multiply $x$ by $x^{2}$ .

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Step 3.4.6.8.2.1

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

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 18 x^{2} + 15 x - 1 \cdot 6 (x^{1} x^{2}) - x^{2} \cdot - 5$

Step 3.4.6.8.2.2

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

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 18 x^{2} + 15 x - 1 \cdot 6 x^{1 + 2} - x^{2} \cdot - 5$

Step 3.4.6.8.3

Add $1$ and $2$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 18 x^{2} + 15 x - 1 \cdot 6 x^{3} - x^{2} \cdot - 5$

Step 3.4.6.9

Multiply $- 1$ by $6$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 18 x^{2} + 15 x - 6 x^{3} - x^{2} \cdot - 5$

Step 3.4.6.10

Multiply $- 5$ by $- 1$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 12 x - 10 - 18 x^{2} + 15 x - 6 x^{3} + 5 x^{2}$

Step 3.4.7

Add $12 x$ and $15 x$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 27 x - 10 - 18 x^{2} - 6 x^{3} + 5 x^{2}$

Step 3.4.8

Add $- 18 x^{2}$ and $5 x^{2}$ .

$x^{2} + 13 x - 3 - 6 x^{3} + 27 x - 10 - 6 x^{3} - 13 x^{2}$

Step 3.5

Subtract $13 x^{2}$ from $x^{2}$ .

$- 12 x^{2} + 13 x - 3 - 6 x^{3} + 27 x - 10 - 6 x^{3}$

Step 3.6

Add $13 x$ and $27 x$ .

$- 12 x^{2} + 40 x - 3 - 6 x^{3} - 10 - 6 x^{3}$

Step 3.7

Subtract $10$ from $- 3$ .

$- 12 x^{2} + 40 x - 6 x^{3} - 13 - 6 x^{3}$

Step 3.8

Subtract $6 x^{3}$ from $- 6 x^{3}$ .

$- 12 x^{2} + 40 x - 12 x^{3} - 13$

Step 4

Evaluate the derivative at $x = 4$ .

$- 12 {(4)}^{2} + 40 (4) - 12 {(4)}^{3} - 13$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Raise $4$ to the power of $2$ .

$- 12 \cdot 16 + 40 (4) - 12 {(4)}^{3} - 13$

Step 5.1.2

Multiply $- 12$ by $16$ .

$- 192 + 40 (4) - 12 {(4)}^{3} - 13$

Step 5.1.3

Multiply $40$ by $4$ .

$- 192 + 160 - 12 {(4)}^{3} - 13$

Step 5.1.4

Raise $4$ to the power of $3$ .

$- 192 + 160 - 12 \cdot 64 - 13$

Step 5.1.5

Multiply $- 12$ by $64$ .

$- 192 + 160 - 768 - 13$

Step 5.2

Simplify by adding and subtracting.

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Step 5.2.1

Add $- 192$ and $160$ .

$- 32 - 768 - 13$

Step 5.2.2

Subtract $768$ from $- 32$ .

$- 800 - 13$

Step 5.2.3

Subtract $13$ from $- 800$ .

$- 813$