Calculus Examples

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

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

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

Step 2

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

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

Step 3

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $3$ by $2$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply $2$ by $- 1$ .

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

Step 5

Differentiate.

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Step 5.1

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

$(3 x - 5) (6 x^{2} - 2 x + 0) + (2 x^{3} - x^{2} + 1) \frac{d}{d x} [3 x - 5]$

Step 5.2

Add $6 x^{2} - 2 x$ and $0$ .

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

Step 5.3

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Multiply $3$ by $1$ .

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

Step 7

Differentiate using the Constant Rule.

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Step 7.1

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

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

Step 7.2

Add $3$ and $0$ .

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Apply the distributive property.

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

Step 8.4

Apply the distributive property.

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

Step 8.5

Combine terms.

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Step 8.5.1

Multiply $6$ by $3$ .

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

Step 8.5.2

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

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Step 8.5.2.1

Move $x^{2}$ .

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

Step 8.5.2.2

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

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Step 8.5.2.2.1

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

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

Step 8.5.2.2.2

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

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

Step 8.5.2.3

Add $2$ and $1$ .

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

Step 8.5.3

Multiply $6$ by $- 5$ .

$18 x^{3} - 30 x^{2} + 3 x (- 2 x) - 5 (- 2 x) + 2 x^{3} \cdot 3 - x^{2} \cdot 3 + 1 \cdot 3$

Step 8.5.4

Multiply $- 2$ by $3$ .

$18 x^{3} - 30 x^{2} - 6 x \cdot x - 5 (- 2 x) + 2 x^{3} \cdot 3 - x^{2} \cdot 3 + 1 \cdot 3$

Step 8.5.5

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

$18 x^{3} - 30 x^{2} - 6 (x^{1} x) - 5 (- 2 x) + 2 x^{3} \cdot 3 - x^{2} \cdot 3 + 1 \cdot 3$

Step 8.5.6

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

$18 x^{3} - 30 x^{2} - 6 (x^{1} x^{1}) - 5 (- 2 x) + 2 x^{3} \cdot 3 - x^{2} \cdot 3 + 1 \cdot 3$

Step 8.5.7

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

$18 x^{3} - 30 x^{2} - 6 x^{1 + 1} - 5 (- 2 x) + 2 x^{3} \cdot 3 - x^{2} \cdot 3 + 1 \cdot 3$

Step 8.5.8

Add $1$ and $1$ .

$18 x^{3} - 30 x^{2} - 6 x^{2} - 5 (- 2 x) + 2 x^{3} \cdot 3 - x^{2} \cdot 3 + 1 \cdot 3$

Step 8.5.9

Multiply $- 2$ by $- 5$ .

$18 x^{3} - 30 x^{2} - 6 x^{2} + 10 x + 2 x^{3} \cdot 3 - x^{2} \cdot 3 + 1 \cdot 3$

Step 8.5.10

Subtract $6 x^{2}$ from $- 30 x^{2}$ .

$18 x^{3} - 36 x^{2} + 10 x + 2 x^{3} \cdot 3 - x^{2} \cdot 3 + 1 \cdot 3$

Step 8.5.11

Multiply $3$ by $2$ .

$18 x^{3} - 36 x^{2} + 10 x + 6 x^{3} - x^{2} \cdot 3 + 1 \cdot 3$

Step 8.5.12

Multiply $3$ by $- 1$ .

$18 x^{3} - 36 x^{2} + 10 x + 6 x^{3} - 3 x^{2} + 1 \cdot 3$

Step 8.5.13

Multiply $3$ by $1$ .

$18 x^{3} - 36 x^{2} + 10 x + 6 x^{3} - 3 x^{2} + 3$

Step 8.5.14

Add $18 x^{3}$ and $6 x^{3}$ .

$24 x^{3} - 36 x^{2} + 10 x - 3 x^{2} + 3$

Step 8.5.15

Subtract $3 x^{2}$ from $- 36 x^{2}$ .

$24 x^{3} - 39 x^{2} + 10 x + 3$