Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

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

Step 2.3

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

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

Step 2.4

Multiply $3$ by $5$ .

$(3 x - 3) (15 x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [1]) + (5 x^{3} - x^{2} + 1) \frac{d}{d x} [3 x - 3]$

Step 2.5

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

$(3 x - 3) (15 x^{2} - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) + (5 x^{3} - x^{2} + 1) \frac{d}{d x} [3 x - 3]$

Step 2.6

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

$(3 x - 3) (15 x^{2} - (2 x) + \frac{d}{d x} [1]) + (5 x^{3} - x^{2} + 1) \frac{d}{d x} [3 x - 3]$

Step 2.7

Multiply $2$ by $- 1$ .

$(3 x - 3) (15 x^{2} - 2 x + \frac{d}{d x} [1]) + (5 x^{3} - x^{2} + 1) \frac{d}{d x} [3 x - 3]$

Step 2.8

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

$(3 x - 3) (15 x^{2} - 2 x + 0) + (5 x^{3} - x^{2} + 1) \frac{d}{d x} [3 x - 3]$

Step 2.9

Add $15 x^{2} - 2 x$ and $0$ .

$(3 x - 3) (15 x^{2} - 2 x) + (5 x^{3} - x^{2} + 1) \frac{d}{d x} [3 x - 3]$

Step 2.10

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

$(3 x - 3) (15 x^{2} - 2 x) + (5 x^{3} - x^{2} + 1) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 3])$

Step 2.11

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

$(3 x - 3) (15 x^{2} - 2 x) + (5 x^{3} - x^{2} + 1) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 2.12

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

$(3 x - 3) (15 x^{2} - 2 x) + (5 x^{3} - x^{2} + 1) (3 \cdot 1 + \frac{d}{d x} [- 3])$

Step 2.13

Multiply $3$ by $1$ .

$(3 x - 3) (15 x^{2} - 2 x) + (5 x^{3} - x^{2} + 1) (3 + \frac{d}{d x} [- 3])$

Step 2.14

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$(3 x - 3) (15 x^{2} - 2 x) + (5 x^{3} - x^{2} + 1) (3 + 0)$

Step 2.15

Simplify the expression.

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Step 2.15.1

Add $3$ and $0$ .

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

Step 2.15.2

Move $3$ to the left of $5 x^{3} - x^{2} + 1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Combine terms.

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Step 3.5.1

Multiply $15$ by $3$ .

$45 x \cdot x^{2} - 3 (15 x^{2}) + 3 x (- 2 x) - 3 (- 2 x) + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.2

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

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Step 3.5.2.1

Move $x^{2}$ .

$45 (x^{2} x) - 3 (15 x^{2}) + 3 x (- 2 x) - 3 (- 2 x) + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.2.2

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

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Step 3.5.2.2.1

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

$45 (x^{2} x^{1}) - 3 (15 x^{2}) + 3 x (- 2 x) - 3 (- 2 x) + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.2.2.2

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

$45 x^{2 + 1} - 3 (15 x^{2}) + 3 x (- 2 x) - 3 (- 2 x) + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.2.3

Add $2$ and $1$ .

$45 x^{3} - 3 (15 x^{2}) + 3 x (- 2 x) - 3 (- 2 x) + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.3

Multiply $15$ by $- 3$ .

$45 x^{3} - 45 x^{2} + 3 x (- 2 x) - 3 (- 2 x) + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.4

Multiply $- 2$ by $3$ .

$45 x^{3} - 45 x^{2} - 6 x \cdot x - 3 (- 2 x) + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.5

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

$45 x^{3} - 45 x^{2} - 6 (x^{1} x) - 3 (- 2 x) + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.6

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

$45 x^{3} - 45 x^{2} - 6 (x^{1} x^{1}) - 3 (- 2 x) + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.7

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

$45 x^{3} - 45 x^{2} - 6 x^{1 + 1} - 3 (- 2 x) + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.8

Add $1$ and $1$ .

$45 x^{3} - 45 x^{2} - 6 x^{2} - 3 (- 2 x) + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.9

Multiply $- 2$ by $- 3$ .

$45 x^{3} - 45 x^{2} - 6 x^{2} + 6 x + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.10

Subtract $6 x^{2}$ from $- 45 x^{2}$ .

$45 x^{3} - 51 x^{2} + 6 x + 3 (5 x^{3}) + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.11

Multiply $5$ by $3$ .

$45 x^{3} - 51 x^{2} + 6 x + 15 x^{3} + 3 (- x^{2}) + 3 \cdot 1$

Step 3.5.12

Multiply $- 1$ by $3$ .

$45 x^{3} - 51 x^{2} + 6 x + 15 x^{3} - 3 x^{2} + 3 \cdot 1$

Step 3.5.13

Multiply $3$ by $1$ .

$45 x^{3} - 51 x^{2} + 6 x + 15 x^{3} - 3 x^{2} + 3$

Step 3.5.14

Add $45 x^{3}$ and $15 x^{3}$ .

$60 x^{3} - 51 x^{2} + 6 x - 3 x^{2} + 3$

Step 3.5.15

Subtract $3 x^{2}$ from $- 51 x^{2}$ .

$60 x^{3} - 54 x^{2} + 6 x + 3$