Calculus Examples

$f (x) = x^{3} (3 x^{2} - 20)$

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 $f (x) = x^{3}$ and $g (x) = 3 x^{2} - 20$ .

$x^{3} \frac{d}{d x} [3 x^{2} - 20] + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 x^{2} - 20$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 20]$ .

$x^{3} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 20]) + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 2.2

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

$x^{3} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 20]) + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 2.3

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

$x^{3} (3 (2 x) + \frac{d}{d x} [- 20]) + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 2.4

Multiply $2$ by $3$ .

$x^{3} (6 x + \frac{d}{d x} [- 20]) + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 2.5

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20$ with respect to $x$ is $0$ .

$x^{3} (6 x + 0) + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 2.6

Add $6 x$ and $0$ .

$x^{3} (6 x) + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 3.1

Move $x$ .

$x \cdot x^{3} \cdot 6 + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 3.2

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

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Step 3.2.1

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

$x^{1} x^{3} \cdot 6 + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 3.2.2

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

$x^{1 + 3} \cdot 6 + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 3.3

Add $1$ and $3$ .

$x^{4} \cdot 6 + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 4

Move $6$ to the left of $x^{4}$ .

$6 \cdot x^{4} + (3 x^{2} - 20) \frac{d}{d x} [x^{3}]$

Step 5

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

$6 x^{4} + (3 x^{2} - 20) (3 x^{2})$

Step 6

Move $3$ to the left of $3 x^{2} - 20$ .

$6 x^{4} + 3 \cdot (3 x^{2} - 20) x^{2}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$6 x^{4} + (3 (3 x^{2}) + 3 \cdot - 20) x^{2}$

Step 7.2

Apply the distributive property.

$6 x^{4} + 3 (3 x^{2}) x^{2} + 3 \cdot - 20 x^{2}$

Step 7.3

Combine terms.

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Step 7.3.1

Multiply $3$ by $3$ .

$6 x^{4} + 9 x^{2} x^{2} + 3 \cdot - 20 x^{2}$

Step 7.3.2

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

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Step 7.3.2.1

Move $x^{2}$ .

$6 x^{4} + 9 (x^{2} x^{2}) + 3 \cdot - 20 x^{2}$

Step 7.3.2.2

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

$6 x^{4} + 9 x^{2 + 2} + 3 \cdot - 20 x^{2}$

Step 7.3.2.3

Add $2$ and $2$ .

$6 x^{4} + 9 x^{4} + 3 \cdot - 20 x^{2}$

Step 7.3.3

Multiply $3$ by $- 20$ .

$6 x^{4} + 9 x^{4} - 60 x^{2}$

Step 7.3.4

Add $6 x^{4}$ and $9 x^{4}$ .

$15 x^{4} - 60 x^{2}$