Calculus Examples

$2 x^{2} (3 x - 4)$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{2} (3 x - 4)$ with respect to $x$ is $2 \frac{d}{d x} [x^{2} (3 x - 4)]$ .

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

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$2 (x^{2} (3 \cdot 1 + \frac{d}{d x} [- 4]) + (3 x - 4) \frac{d}{d x} [x^{2}])$

Step 3.4

Multiply $3$ by $1$ .

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

Step 3.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

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

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $3$ and $0$ .

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

Step 3.6.2

Move $3$ to the left of $x^{2}$ .

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

Step 3.7

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

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

Step 3.8

Move $2$ to the left of $3 x - 4$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Combine terms.

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Step 4.4.1

Multiply $3$ by $2$ .

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

Step 4.4.2

Multiply $3$ by $2$ .

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

Step 4.4.3

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

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

Step 4.4.4

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

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

Step 4.4.5

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

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

Step 4.4.6

Add $1$ and $1$ .

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

Step 4.4.7

Multiply $6$ by $2$ .

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

Step 4.4.8

Multiply $2$ by $- 4$ .

$6 x^{2} + 12 x^{2} + 2 (- 8 x)$

Step 4.4.9

Multiply $- 8$ by $2$ .

$6 x^{2} + 12 x^{2} - 16 x$

Step 4.4.10

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

$18 x^{2} - 16 x$