Calculus Examples

$y = 3 x (x^{3} - 1)$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 x (x^{3} - 1)$ with respect to $x$ is $3 \frac{d}{d x} [x (x^{3} - 1)]$ .

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $3 x^{2}$ and $0$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $2$ .

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

Step 7

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

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

Step 8

Simplify by adding terms.

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Step 8.1

Multiply $x^{3} - 1$ by $1$ .

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

Step 8.2

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

$3 (4 x^{3} - 1)$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$3 (4 x^{3}) + 3 \cdot - 1$

Step 9.2

Combine terms.

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Step 9.2.1

Multiply $4$ by $3$ .

$12 x^{3} + 3 \cdot - 1$

Step 9.2.2

Multiply $3$ by $- 1$ .

$12 x^{3} - 3$