Calculus Examples

$y = (5 x^{2} - 1) (3 x^{3} + x)$

Step 1

Find the first derivative.

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Step 1.1

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 $f (x) = 5 x^{2} - 1$ and $g (x) = 3 x^{3} + x$ .

$f' (x) = (5 x^{2} - 1) \frac{d}{d x} (3 x^{3} + x) + (3 x^{3} + x) \frac{d}{d x} (5 x^{2} - 1)$

Step 1.2

Differentiate.

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Step 1.2.1

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

$f' (x) = (5 x^{2} - 1) (\frac{d}{d x} (3 x^{3}) + \frac{d}{d x} (x)) + (3 x^{3} + x) \frac{d}{d x} (5 x^{2} - 1)$

Step 1.2.2

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

$f' (x) = (5 x^{2} - 1) (3 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (x)) + (3 x^{3} + x) \frac{d}{d x} (5 x^{2} - 1)$

Step 1.2.3

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

$f' (x) = (5 x^{2} - 1) (3 (3 x^{2}) + \frac{d}{d x} (x)) + (3 x^{3} + x) \frac{d}{d x} (5 x^{2} - 1)$

Step 1.2.4

Multiply $3$ by $3$ .

$f' (x) = (5 x^{2} - 1) (9 x^{2} + \frac{d}{d x} (x)) + (3 x^{3} + x) \frac{d}{d x} (5 x^{2} - 1)$

Step 1.2.5

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

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + (3 x^{3} + x) \frac{d}{d x} (5 x^{2} - 1)$

Step 1.2.6

By the Sum Rule, the derivative of $5 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 1]$ .

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + (3 x^{3} + x) (\frac{d}{d x} (5 x^{2}) + \frac{d}{d x} (- 1))$

Step 1.2.7

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + (3 x^{3} + x) (5 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 1))$

Step 1.2.8

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

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + (3 x^{3} + x) (5 (2 x) + \frac{d}{d x} (- 1))$

Step 1.2.9

Multiply $2$ by $5$ .

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + (3 x^{3} + x) (10 x + \frac{d}{d x} (- 1))$

Step 1.2.10

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

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + (3 x^{3} + x) (10 x + 0)$

Step 1.2.11

Simplify the expression.

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Step 1.2.11.1

Add $10 x$ and $0$ .

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + (3 x^{3} + x) (10 x)$

Step 1.2.11.2

Move $10$ to the left of $3 x^{3} + x$ .

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + 10 \cdot ((3 x^{3} + x) x)$

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + 10 \cdot ((3 x^{3} + x) x)$

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + 10 \cdot ((3 x^{3} + x) x)$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + (10 (3 x^{3}) + 10 x) x$

Step 1.3.2

Apply the distributive property.

$f' (x) = (5 x^{2} - 1) (9 x^{2} + 1) + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.3

Apply the distributive property.

$f' (x) = (5 x^{2} - 1) (9 x^{2}) + (5 x^{2} - 1) \cdot 1 + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.4

Apply the distributive property.

$f' (x) = 5 x^{2} (9 x^{2}) - 1 (9 x^{2}) + (5 x^{2} - 1) \cdot 1 + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.5

Apply the distributive property.

$f' (x) = 5 x^{2} (9 x^{2}) - 1 (9 x^{2}) + 5 x^{2} \cdot 1 - 1 \cdot 1 + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.6

Combine terms.

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Step 1.3.6.1

Multiply $9$ by $5$ .

$f' (x) = 45 x^{2} x^{2} - 1 (9 x^{2}) + 5 x^{2} \cdot 1 - 1 \cdot 1 + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.6.2

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

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Step 1.3.6.2.1

Move $x^{2}$ .

$f' (x) = 45 (x^{2} x^{2}) - 1 (9 x^{2}) + 5 x^{2} \cdot 1 - 1 \cdot 1 + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.6.2.2

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

$f' (x) = 45 x^{2 + 2} - 1 (9 x^{2}) + 5 x^{2} \cdot 1 - 1 \cdot 1 + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.6.2.3

Add $2$ and $2$ .

$f' (x) = 45 x^{4} - 1 (9 x^{2}) + 5 x^{2} \cdot 1 - 1 \cdot 1 + 10 (3 x^{3}) x + 10 x \cdot x$

$f' (x) = 45 x^{4} - 1 (9 x^{2}) + 5 x^{2} \cdot 1 - 1 \cdot 1 + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.6.3

Multiply $9$ by $- 1$ .

$f' (x) = 45 x^{4} - 9 x^{2} + 5 x^{2} \cdot 1 - 1 \cdot 1 + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.6.4

Multiply $5$ by $1$ .

$f' (x) = 45 x^{4} - 9 x^{2} + 5 x^{2} - 1 \cdot 1 + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.6.5

Multiply $- 1$ by $1$ .

$f' (x) = 45 x^{4} - 9 x^{2} + 5 x^{2} - 1 + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.6.6

Add $- 9 x^{2}$ and $5 x^{2}$ .

$f' (x) = 45 x^{4} - 4 x^{2} - 1 + 10 (3 x^{3}) x + 10 x \cdot x$

Step 1.3.6.7

Multiply $3$ by $10$ .

$f' (x) = 45 x^{4} - 4 x^{2} - 1 + 30 x^{3} x + 10 x \cdot x$

Step 1.3.6.8

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

$f' (x) = 45 x^{4} - 4 x^{2} - 1 + 30 (x \cdot x^{3}) + 10 x \cdot x$

Step 1.3.6.9

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

$f' (x) = 45 x^{4} - 4 x^{2} - 1 + 30 x^{1 + 3} + 10 x \cdot x$

Step 1.3.6.10

Add $1$ and $3$ .

$f' (x) = 45 x^{4} - 4 x^{2} - 1 + 30 x^{4} + 10 x \cdot x$

Step 1.3.6.11

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

$f' (x) = 45 x^{4} - 4 x^{2} - 1 + 30 x^{4} + 10 (x \cdot x)$

Step 1.3.6.12

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

$f' (x) = 45 x^{4} - 4 x^{2} - 1 + 30 x^{4} + 10 (x \cdot x)$

Step 1.3.6.13

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

$f' (x) = 45 x^{4} - 4 x^{2} - 1 + 30 x^{4} + 10 x^{1 + 1}$

Step 1.3.6.14

Add $1$ and $1$ .

$f' (x) = 45 x^{4} - 4 x^{2} - 1 + 30 x^{4} + 10 x^{2}$

Step 1.3.6.15

Add $45 x^{4}$ and $30 x^{4}$ .

$f' (x) = 75 x^{4} - 4 x^{2} - 1 + 10 x^{2}$

Step 1.3.6.16

Add $- 4 x^{2}$ and $10 x^{2}$ .

$f' (x) = 75 x^{4} + 6 x^{2} - 1$

$f' (x) = 75 x^{4} + 6 x^{2} - 1$

$f' (x) = 75 x^{4} + 6 x^{2} - 1$

$f' (x) = 75 x^{4} + 6 x^{2} - 1$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $75 x^{4} + 6 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [75 x^{4}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 1]$ .

$f'' (x) = \frac{d}{d x} (75 x^{4}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (- 1)$

Step 2.2

Evaluate $\frac{d}{d x} [75 x^{4}]$ .

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Step 2.2.1

Since $75$ is constant with respect to $x$ , the derivative of $75 x^{4}$ with respect to $x$ is $75 \frac{d}{d x} [x^{4}]$ .

$f'' (x) = 75 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (- 1)$

Step 2.2.2

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

$f'' (x) = 75 (4 x^{3}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (- 1)$

Step 2.2.3

Multiply $4$ by $75$ .

$f'' (x) = 300 x^{3} + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (- 1)$

$f'' (x) = 300 x^{3} + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (- 1)$

Step 2.3

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 2.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 300 x^{3} + 6 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 1)$

Step 2.3.2

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

$f'' (x) = 300 x^{3} + 6 (2 x) + \frac{d}{d x} (- 1)$

Step 2.3.3

Multiply $2$ by $6$ .

$f'' (x) = 300 x^{3} + 12 x + \frac{d}{d x} (- 1)$

$f'' (x) = 300 x^{3} + 12 x + \frac{d}{d x} (- 1)$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

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

$f'' (x) = 300 x^{3} + 12 x + 0$

Step 2.4.2

Add $300 x^{3} + 12 x$ and $0$ .

$f'' (x) = 300 x^{3} + 12 x$

$f'' (x) = 300 x^{3} + 12 x$

$f'' (x) = 300 x^{3} + 12 x$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $300 x^{3} + 12 x$ with respect to $x$ is $\frac{d}{d x} [300 x^{3}] + \frac{d}{d x} [12 x]$ .

$f''' (x) = \frac{d}{d x} (300 x^{3}) + \frac{d}{d x} (12 x)$

Step 3.2

Evaluate $\frac{d}{d x} [300 x^{3}]$ .

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Step 3.2.1

Since $300$ is constant with respect to $x$ , the derivative of $300 x^{3}$ with respect to $x$ is $300 \frac{d}{d x} [x^{3}]$ .

$f''' (x) = 300 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (12 x)$

Step 3.2.2

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

$f''' (x) = 300 (3 x^{2}) + \frac{d}{d x} (12 x)$

Step 3.2.3

Multiply $3$ by $300$ .

$f''' (x) = 900 x^{2} + \frac{d}{d x} (12 x)$

$f''' (x) = 900 x^{2} + \frac{d}{d x} (12 x)$

Step 3.3

Evaluate $\frac{d}{d x} [12 x]$ .

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Step 3.3.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$f''' (x) = 900 x^{2} + 12 \frac{d}{d x} (x)$

Step 3.3.2

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

$f''' (x) = 900 x^{2} + 12 \cdot 1$

Step 3.3.3

Multiply $12$ by $1$ .

$f''' (x) = 900 x^{2} + 12$

$f''' (x) = 900 x^{2} + 12$

$f''' (x) = 900 x^{2} + 12$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $900 x^{2} + 12$ with respect to $x$ is $\frac{d}{d x} [900 x^{2}] + \frac{d}{d x} [12]$ .

$f^{4} (x) = \frac{d}{d x} (900 x^{2}) + \frac{d}{d x} (12)$

Step 4.2

Evaluate $\frac{d}{d x} [900 x^{2}]$ .

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Step 4.2.1

Since $900$ is constant with respect to $x$ , the derivative of $900 x^{2}$ with respect to $x$ is $900 \frac{d}{d x} [x^{2}]$ .

$f^{4} (x) = 900 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (12)$

Step 4.2.2

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

$f^{4} (x) = 900 (2 x) + \frac{d}{d x} (12)$

Step 4.2.3

Multiply $2$ by $900$ .

$f^{4} (x) = 1800 x + \frac{d}{d x} (12)$

$f^{4} (x) = 1800 x + \frac{d}{d x} (12)$

Step 4.3

Differentiate using the Constant Rule.

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Step 4.3.1

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$f^{4} (x) = 1800 x + 0$

Step 4.3.2

Add $1800 x$ and $0$ .

$f^{4} (x) = 1800 x$

$f^{4} (x) = 1800 x$

$f^{4} (x) = 1800 x$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$