Calculus Examples

$y = x (4 x^{3} + 3 x^{2} - 6)$

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 $f (x) = x$ and $g (x) = 4 x^{3} + 3 x^{2} - 6$ .

$x \frac{d}{d x} [4 x^{3} + 3 x^{2} - 6] + (4 x^{3} + 3 x^{2} - 6) \frac{d}{d x} [x]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 x^{3} + 3 x^{2} - 6$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6]$ .

$x (\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6]) + (4 x^{3} + 3 x^{2} - 6) \frac{d}{d x} [x]$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$x (4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6]) + (4 x^{3} + 3 x^{2} - 6) \frac{d}{d x} [x]$

Step 2.3

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

$x (4 (3 x^{2}) + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6]) + (4 x^{3} + 3 x^{2} - 6) \frac{d}{d x} [x]$

Step 2.4

Multiply $3$ by $4$ .

$x (12 x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 6]) + (4 x^{3} + 3 x^{2} - 6) \frac{d}{d x} [x]$

Step 2.5

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

$x (12 x^{2} + 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6]) + (4 x^{3} + 3 x^{2} - 6) \frac{d}{d x} [x]$

Step 2.6

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

$x (12 x^{2} + 3 (2 x) + \frac{d}{d x} [- 6]) + (4 x^{3} + 3 x^{2} - 6) \frac{d}{d x} [x]$

Step 2.7

Multiply $2$ by $3$ .

$x (12 x^{2} + 6 x + \frac{d}{d x} [- 6]) + (4 x^{3} + 3 x^{2} - 6) \frac{d}{d x} [x]$

Step 2.8

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$x (12 x^{2} + 6 x + 0) + (4 x^{3} + 3 x^{2} - 6) \frac{d}{d x} [x]$

Step 2.9

Add $12 x^{2} + 6 x$ and $0$ .

$x (12 x^{2} + 6 x) + (4 x^{3} + 3 x^{2} - 6) \frac{d}{d x} [x]$

Step 2.10

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

$x (12 x^{2} + 6 x) + (4 x^{3} + 3 x^{2} - 6) \cdot 1$

Step 2.11

Multiply $4 x^{3} + 3 x^{2} - 6$ by $1$ .

$x (12 x^{2} + 6 x) + 4 x^{3} + 3 x^{2} - 6$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$x (12 x^{2}) + x (6 x) + 4 x^{3} + 3 x^{2} - 6$

Step 3.2

Combine terms.

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Step 3.2.1

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

$12 (x^{1} x^{2}) + x (6 x) + 4 x^{3} + 3 x^{2} - 6$

Step 3.2.2

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

$12 x^{1 + 2} + x (6 x) + 4 x^{3} + 3 x^{2} - 6$

Step 3.2.3

Add $1$ and $2$ .

$12 x^{3} + x (6 x) + 4 x^{3} + 3 x^{2} - 6$

Step 3.2.4

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

$12 x^{3} + 6 (x^{1} x) + 4 x^{3} + 3 x^{2} - 6$

Step 3.2.5

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

$12 x^{3} + 6 (x^{1} x^{1}) + 4 x^{3} + 3 x^{2} - 6$

Step 3.2.6

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

$12 x^{3} + 6 x^{1 + 1} + 4 x^{3} + 3 x^{2} - 6$

Step 3.2.7

Add $1$ and $1$ .

$12 x^{3} + 6 x^{2} + 4 x^{3} + 3 x^{2} - 6$

Step 3.2.8

Add $12 x^{3}$ and $4 x^{3}$ .

$16 x^{3} + 6 x^{2} + 3 x^{2} - 6$

Step 3.2.9

Add $6 x^{2}$ and $3 x^{2}$ .

$16 x^{3} + 9 x^{2} - 6$