Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Add $2 x$ and $0$ .

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

Step 1.4

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

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Step 1.4.1

Move $x$ .

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

Step 1.4.2

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

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Step 1.4.2.1

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

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

Step 1.4.2.2

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

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

Step 1.4.3

Add $1$ and $3$ .

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

Step 1.5

Move $2$ to the left of $x^{4}$ .

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

Simplify.

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Step 1.8.1

Apply the distributive property.

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

Step 1.8.2

Apply the distributive property.

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

Step 1.8.3

Apply the distributive property.

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

Step 1.8.4

Combine terms.

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Step 1.8.4.1

Multiply $2$ by $3$ .

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

Step 1.8.4.2

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

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Step 1.8.4.2.1

Move $x^{2}$ .

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

Step 1.8.4.2.2

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

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

Step 1.8.4.2.3

Add $2$ and $2$ .

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

Step 1.8.4.3

Multiply $3$ by $3$ .

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

Step 1.8.4.4

Multiply $3$ by $- 4$ .

$6 x^{4} + 9 x^{4} + 3 (- 12 x^{2})$

Step 1.8.4.5

Multiply $- 12$ by $3$ .

$6 x^{4} + 9 x^{4} - 36 x^{2}$

Step 1.8.4.6

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

$f' (x) = 15 x^{4} - 36 x^{2}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $15 x^{4} - 36 x^{2}$ with respect to $x$ is $\frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [- 36 x^{2}]$ .

$\frac{d}{d x} [15 x^{4}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2

Evaluate $\frac{d}{d x} [15 x^{4}]$ .

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Step 2.2.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{4}$ with respect to $x$ is $15 \frac{d}{d x} [x^{4}]$ .

$15 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$15 (4 x^{3}) + \frac{d}{d x} [- 36 x^{2}]$

Step 2.2.3

Multiply $4$ by $15$ .

$60 x^{3} + \frac{d}{d x} [- 36 x^{2}]$

Step 2.3

Evaluate $\frac{d}{d x} [- 36 x^{2}]$ .

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Step 2.3.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 x^{2}$ with respect to $x$ is $- 36 \frac{d}{d x} [x^{2}]$ .

$60 x^{3} - 36 \frac{d}{d x} [x^{2}]$

Step 2.3.2

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

$60 x^{3} - 36 (2 x)$

Step 2.3.3

Multiply $2$ by $- 36$ .

$f'' (x) = 60 x^{3} - 72 x$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $60 x^{3} - 72 x$ with respect to $x$ is $\frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [- 72 x]$ .

$\frac{d}{d x} [60 x^{3}] + \frac{d}{d x} [- 72 x]$

Step 3.2

Evaluate $\frac{d}{d x} [60 x^{3}]$ .

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Step 3.2.1

Since $60$ is constant with respect to $x$ , the derivative of $60 x^{3}$ with respect to $x$ is $60 \frac{d}{d x} [x^{3}]$ .

$60 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 72 x]$

Step 3.2.2

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

$60 (3 x^{2}) + \frac{d}{d x} [- 72 x]$

Step 3.2.3

Multiply $3$ by $60$ .

$180 x^{2} + \frac{d}{d x} [- 72 x]$

Step 3.3

Evaluate $\frac{d}{d x} [- 72 x]$ .

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Step 3.3.1

Since $- 72$ is constant with respect to $x$ , the derivative of $- 72 x$ with respect to $x$ is $- 72 \frac{d}{d x} [x]$ .

$180 x^{2} - 72 \frac{d}{d x} [x]$

Step 3.3.2

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

$180 x^{2} - 72 \cdot 1$

Step 3.3.3

Multiply $- 72$ by $1$ .

$f''' (x) = 180 x^{2} - 72$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $180 x^{2} - 72$ with respect to $x$ is $\frac{d}{d x} [180 x^{2}] + \frac{d}{d x} [- 72]$ .

$\frac{d}{d x} [180 x^{2}] + \frac{d}{d x} [- 72]$

Step 4.2

Evaluate $\frac{d}{d x} [180 x^{2}]$ .

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Step 4.2.1

Since $180$ is constant with respect to $x$ , the derivative of $180 x^{2}$ with respect to $x$ is $180 \frac{d}{d x} [x^{2}]$ .

$180 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 72]$

Step 4.2.2

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

$180 (2 x) + \frac{d}{d x} [- 72]$

Step 4.2.3

Multiply $2$ by $180$ .

$360 x + \frac{d}{d x} [- 72]$

Step 4.3

Differentiate using the Constant Rule.

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Step 4.3.1

Since $- 72$ is constant with respect to $x$ , the derivative of $- 72$ with respect to $x$ is $0$ .

$360 x + 0$

Step 4.3.2

Add $360 x$ and $0$ .

$f^{4} (x) = 360 x$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $360 x$ .

$360 x$