Calculus Examples

$y = (7 x^{2} - 1) (2 x^{3} + x)$

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 $f (x) = 7 x^{2} - 1$ and $g (x) = 2 x^{3} + x$ .

$f' (x) = (7 x^{2} - 1) \frac{d}{d x} (2 x^{3} + x) + (2 x^{3} + x) \frac{d}{d x} (7 x^{2} - 1)$

Step 1.2

Differentiate.

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Step 1.2.1

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

$f' (x) = (7 x^{2} - 1) (\frac{d}{d x} (2 x^{3}) + \frac{d}{d x} (x)) + (2 x^{3} + x) \frac{d}{d x} (7 x^{2} - 1)$

Step 1.2.2

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

$f' (x) = (7 x^{2} - 1) (2 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (x)) + (2 x^{3} + x) \frac{d}{d x} (7 x^{2} - 1)$

Step 1.2.3

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

$f' (x) = (7 x^{2} - 1) (2 (3 x^{2}) + \frac{d}{d x} (x)) + (2 x^{3} + x) \frac{d}{d x} (7 x^{2} - 1)$

Step 1.2.4

Multiply $3$ by $2$ .

$f' (x) = (7 x^{2} - 1) (6 x^{2} + \frac{d}{d x} (x)) + (2 x^{3} + x) \frac{d}{d x} (7 x^{2} - 1)$

Step 1.2.5

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

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + (2 x^{3} + x) \frac{d}{d x} (7 x^{2} - 1)$

Step 1.2.6

By the Sum Rule, the derivative of $7 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [- 1]$ .

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + (2 x^{3} + x) (\frac{d}{d x} (7 x^{2}) + \frac{d}{d x} (- 1))$

Step 1.2.7

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + (2 x^{3} + x) (7 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 1))$

Step 1.2.8

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

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + (2 x^{3} + x) (7 (2 x) + \frac{d}{d x} (- 1))$

Step 1.2.9

Multiply $2$ by $7$ .

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + (2 x^{3} + x) (14 x + \frac{d}{d x} (- 1))$

Step 1.2.10

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

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + (2 x^{3} + x) (14 x + 0)$

Step 1.2.11

Simplify the expression.

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Step 1.2.11.1

Add $14 x$ and $0$ .

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + (2 x^{3} + x) (14 x)$

Step 1.2.11.2

Move $14$ to the left of $2 x^{3} + x$ .

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + 14 \cdot ((2 x^{3} + x) x)$

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + 14 \cdot ((2 x^{3} + x) x)$

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + 14 \cdot ((2 x^{3} + x) x)$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + (14 (2 x^{3}) + 14 x) x$

Step 1.3.2

Apply the distributive property.

$f' (x) = (7 x^{2} - 1) (6 x^{2} + 1) + 14 (2 x^{3}) x + 14 x \cdot x$

Step 1.3.3

Apply the distributive property.

$f' (x) = (7 x^{2} - 1) (6 x^{2}) + (7 x^{2} - 1) \cdot 1 + 14 (2 x^{3}) x + 14 x \cdot x$

Step 1.3.4

Apply the distributive property.

$f' (x) = 7 x^{2} (6 x^{2}) - 1 (6 x^{2}) + (7 x^{2} - 1) \cdot 1 + 14 (2 x^{3}) x + 14 x \cdot x$

Step 1.3.5

Apply the distributive property.

$f' (x) = 7 x^{2} (6 x^{2}) - 1 (6 x^{2}) + 7 x^{2} \cdot 1 - 1 \cdot 1 + 14 (2 x^{3}) x + 14 x \cdot x$

Step 1.3.6

Combine terms.

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Step 1.3.6.1

Multiply $6$ by $7$ .

$f' (x) = 42 x^{2} x^{2} - 1 (6 x^{2}) + 7 x^{2} \cdot 1 - 1 \cdot 1 + 14 (2 x^{3}) x + 14 x \cdot x$

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 $x^{2}$ .

$f' (x) = 42 (x^{2} x^{2}) - 1 (6 x^{2}) + 7 x^{2} \cdot 1 - 1 \cdot 1 + 14 (2 x^{3}) x + 14 x \cdot x$

Step 1.3.6.2.2

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

$f' (x) = 42 x^{2 + 2} - 1 (6 x^{2}) + 7 x^{2} \cdot 1 - 1 \cdot 1 + 14 (2 x^{3}) x + 14 x \cdot x$

Step 1.3.6.2.3

Add $2$ and $2$ .

$f' (x) = 42 x^{4} - 1 (6 x^{2}) + 7 x^{2} \cdot 1 - 1 \cdot 1 + 14 (2 x^{3}) x + 14 x \cdot x$

$f' (x) = 42 x^{4} - 1 (6 x^{2}) + 7 x^{2} \cdot 1 - 1 \cdot 1 + 14 (2 x^{3}) x + 14 x \cdot x$

Step 1.3.6.3

Multiply $6$ by $- 1$ .

$f' (x) = 42 x^{4} - 6 x^{2} + 7 x^{2} \cdot 1 - 1 \cdot 1 + 14 (2 x^{3}) x + 14 x \cdot x$

Step 1.3.6.4

Multiply $7$ by $1$ .

$f' (x) = 42 x^{4} - 6 x^{2} + 7 x^{2} - 1 \cdot 1 + 14 (2 x^{3}) x + 14 x \cdot x$

Step 1.3.6.5

Multiply $- 1$ by $1$ .

$f' (x) = 42 x^{4} - 6 x^{2} + 7 x^{2} - 1 + 14 (2 x^{3}) x + 14 x \cdot x$

Step 1.3.6.6

Add $- 6 x^{2}$ and $7 x^{2}$ .

$f' (x) = 42 x^{4} + x^{2} - 1 + 14 (2 x^{3}) x + 14 x \cdot x$

Step 1.3.6.7

Multiply $2$ by $14$ .

$f' (x) = 42 x^{4} + x^{2} - 1 + 28 x^{3} x + 14 x \cdot x$

Step 1.3.6.8

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

$f' (x) = 42 x^{4} + x^{2} - 1 + 28 (x \cdot x^{3}) + 14 x \cdot x$

Step 1.3.6.9

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

$f' (x) = 42 x^{4} + x^{2} - 1 + 28 x^{1 + 3} + 14 x \cdot x$

Step 1.3.6.10

Add $1$ and $3$ .

$f' (x) = 42 x^{4} + x^{2} - 1 + 28 x^{4} + 14 x \cdot x$

Step 1.3.6.11

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

$f' (x) = 42 x^{4} + x^{2} - 1 + 28 x^{4} + 14 (x \cdot x)$

Step 1.3.6.12

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

$f' (x) = 42 x^{4} + x^{2} - 1 + 28 x^{4} + 14 (x \cdot x)$

Step 1.3.6.13

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

$f' (x) = 42 x^{4} + x^{2} - 1 + 28 x^{4} + 14 x^{1 + 1}$

Step 1.3.6.14

Add $1$ and $1$ .

$f' (x) = 42 x^{4} + x^{2} - 1 + 28 x^{4} + 14 x^{2}$

Step 1.3.6.15

Add $42 x^{4}$ and $28 x^{4}$ .

$f' (x) = 70 x^{4} + x^{2} - 1 + 14 x^{2}$

Step 1.3.6.16

Add $x^{2}$ and $14 x^{2}$ .

$f' (x) = 70 x^{4} + 15 x^{2} - 1$

$f' (x) = 70 x^{4} + 15 x^{2} - 1$

$f' (x) = 70 x^{4} + 15 x^{2} - 1$

$f' (x) = 70 x^{4} + 15 x^{2} - 1$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $70 x^{4} + 15 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [70 x^{4}] + \frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [- 1]$ .

$f'' (x) = \frac{d}{d x} (70 x^{4}) + \frac{d}{d x} (15 x^{2}) + \frac{d}{d x} (- 1)$

Step 2.2

Evaluate $\frac{d}{d x} [70 x^{4}]$ .

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Step 2.2.1

Since $70$ is constant with respect to $x$ , the derivative of $70 x^{4}$ with respect to $x$ is $70 \frac{d}{d x} [x^{4}]$ .

$f'' (x) = 70 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (15 x^{2}) + \frac{d}{d x} (- 1)$

Step 2.2.2

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

$f'' (x) = 70 (4 x^{3}) + \frac{d}{d x} (15 x^{2}) + \frac{d}{d x} (- 1)$

Step 2.2.3

Multiply $4$ by $70$ .

$f'' (x) = 280 x^{3} + \frac{d}{d x} (15 x^{2}) + \frac{d}{d x} (- 1)$

$f'' (x) = 280 x^{3} + \frac{d}{d x} (15 x^{2}) + \frac{d}{d x} (- 1)$

Step 2.3

Evaluate $\frac{d}{d x} [15 x^{2}]$ .

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Step 2.3.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{2}$ with respect to $x$ is $15 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 280 x^{3} + 15 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 1)$

Step 2.3.2

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

$f'' (x) = 280 x^{3} + 15 (2 x) + \frac{d}{d x} (- 1)$

Step 2.3.3

Multiply $2$ by $15$ .

$f'' (x) = 280 x^{3} + 30 x + \frac{d}{d x} (- 1)$

$f'' (x) = 280 x^{3} + 30 x + \frac{d}{d x} (- 1)$

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 $0$ .

$f'' (x) = 280 x^{3} + 30 x + 0$

Step 2.4.2

Add $280 x^{3} + 30 x$ and $0$ .

$f'' (x) = 280 x^{3} + 30 x$

$f'' (x) = 280 x^{3} + 30 x$

$f'' (x) = 280 x^{3} + 30 x$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $280 x^{3} + 30 x$ with respect to $x$ is $\frac{d}{d x} [280 x^{3}] + \frac{d}{d x} [30 x]$ .

$f''' (x) = \frac{d}{d x} (280 x^{3}) + \frac{d}{d x} (30 x)$

Step 3.2

Evaluate $\frac{d}{d x} [280 x^{3}]$ .

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Step 3.2.1

Since $280$ is constant with respect to $x$ , the derivative of $280 x^{3}$ with respect to $x$ is $280 \frac{d}{d x} [x^{3}]$ .

$f''' (x) = 280 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (30 x)$

Step 3.2.2

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

$f''' (x) = 280 (3 x^{2}) + \frac{d}{d x} (30 x)$

Step 3.2.3

Multiply $3$ by $280$ .

$f''' (x) = 840 x^{2} + \frac{d}{d x} (30 x)$

$f''' (x) = 840 x^{2} + \frac{d}{d x} (30 x)$

Step 3.3

Evaluate $\frac{d}{d x} [30 x]$ .

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Step 3.3.1

Since $30$ is constant with respect to $x$ , the derivative of $30 x$ with respect to $x$ is $30 \frac{d}{d x} [x]$ .

$f''' (x) = 840 x^{2} + 30 \frac{d}{d x} (x)$

Step 3.3.2

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

$f''' (x) = 840 x^{2} + 30 \cdot 1$

Step 3.3.3

Multiply $30$ by $1$ .

$f''' (x) = 840 x^{2} + 30$

$f''' (x) = 840 x^{2} + 30$

$f''' (x) = 840 x^{2} + 30$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $840 x^{2} + 30$ with respect to $x$ is $\frac{d}{d x} [840 x^{2}] + \frac{d}{d x} [30]$ .

$f^{4} (x) = \frac{d}{d x} (840 x^{2}) + \frac{d}{d x} (30)$

Step 4.2

Evaluate $\frac{d}{d x} [840 x^{2}]$ .

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Step 4.2.1

Since $840$ is constant with respect to $x$ , the derivative of $840 x^{2}$ with respect to $x$ is $840 \frac{d}{d x} [x^{2}]$ .

$f^{4} (x) = 840 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (30)$

Step 4.2.2

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

$f^{4} (x) = 840 (2 x) + \frac{d}{d x} (30)$

Step 4.2.3

Multiply $2$ by $840$ .

$f^{4} (x) = 1680 x + \frac{d}{d x} (30)$

$f^{4} (x) = 1680 x + \frac{d}{d x} (30)$

Step 4.3

Differentiate using the Constant Rule.

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Step 4.3.1

Since $30$ is constant with respect to $x$ , the derivative of $30$ with respect to $x$ is $0$ .

$f^{4} (x) = 1680 x + 0$

Step 4.3.2

Add $1680 x$ and $0$ .

$f^{4} (x) = 1680 x$

$f^{4} (x) = 1680 x$

$f^{4} (x) = 1680 x$

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