Calculus Examples

$y = (5 x^{2} - 3) (7 x^{3} + x)$

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 $f (x) = 5 x^{2} - 3$ and $g (x) = 7 x^{3} + x$ .

$(5 x^{2} - 3) \frac{d}{d x} [7 x^{3} + x] + (7 x^{3} + x) \frac{d}{d x} [5 x^{2} - 3]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $7 x^{3} + x$ with respect to $x$ is $\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [x]$ .

$(5 x^{2} - 3) (\frac{d}{d x} [7 x^{3}] + \frac{d}{d x} [x]) + (7 x^{3} + x) \frac{d}{d x} [5 x^{2} - 3]$

Step 1.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{3}$ with respect to $x$ is $7 \frac{d}{d x} [x^{3}]$ .

$(5 x^{2} - 3) (7 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x]) + (7 x^{3} + x) \frac{d}{d x} [5 x^{2} - 3]$

Step 1.2.3

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

$(5 x^{2} - 3) (7 (3 x^{2}) + \frac{d}{d x} [x]) + (7 x^{3} + x) \frac{d}{d x} [5 x^{2} - 3]$

Step 1.2.4

Multiply $3$ by $7$ .

$(5 x^{2} - 3) (21 x^{2} + \frac{d}{d x} [x]) + (7 x^{3} + x) \frac{d}{d x} [5 x^{2} - 3]$

Step 1.2.5

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

$(5 x^{2} - 3) (21 x^{2} + 1) + (7 x^{3} + x) \frac{d}{d x} [5 x^{2} - 3]$

Step 1.2.6

By the Sum Rule, the derivative of $5 x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 3]$ .

$(5 x^{2} - 3) (21 x^{2} + 1) + (7 x^{3} + x) (\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 3])$

Step 1.2.7

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

$(5 x^{2} - 3) (21 x^{2} + 1) + (7 x^{3} + x) (5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3])$

Step 1.2.8

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

$(5 x^{2} - 3) (21 x^{2} + 1) + (7 x^{3} + x) (5 (2 x) + \frac{d}{d x} [- 3])$

Step 1.2.9

Multiply $2$ by $5$ .

$(5 x^{2} - 3) (21 x^{2} + 1) + (7 x^{3} + x) (10 x + \frac{d}{d x} [- 3])$

Step 1.2.10

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$(5 x^{2} - 3) (21 x^{2} + 1) + (7 x^{3} + x) (10 x + 0)$

Step 1.2.11

Simplify the expression.

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Step 1.2.11.1

Add $10 x$ and $0$ .

$(5 x^{2} - 3) (21 x^{2} + 1) + (7 x^{3} + x) (10 x)$

Step 1.2.11.2

Move $10$ to the left of $7 x^{3} + x$ .

$(5 x^{2} - 3) (21 x^{2} + 1) + 10 \cdot (7 x^{3} + x) x$

$(5 x^{2} - 3) (21 x^{2} + 1) + 10 \cdot (7 x^{3} + x) x$

$(5 x^{2} - 3) (21 x^{2} + 1) + 10 \cdot (7 x^{3} + x) x$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$(5 x^{2} - 3) (21 x^{2} + 1) + (10 (7 x^{3}) + 10 x) x$

Step 1.3.2

Apply the distributive property.

$(5 x^{2} - 3) (21 x^{2} + 1) + 10 (7 x^{3}) x + 10 x \cdot x$

Step 1.3.3

Apply the distributive property.

$(5 x^{2} - 3) (21 x^{2}) + (5 x^{2} - 3) \cdot 1 + 10 (7 x^{3}) x + 10 x \cdot x$

Step 1.3.4

Apply the distributive property.

$5 x^{2} (21 x^{2}) - 3 (21 x^{2}) + (5 x^{2} - 3) \cdot 1 + 10 (7 x^{3}) x + 10 x \cdot x$

Step 1.3.5

Apply the distributive property.

$5 x^{2} (21 x^{2}) - 3 (21 x^{2}) + 5 x^{2} \cdot 1 - 3 \cdot 1 + 10 (7 x^{3}) x + 10 x \cdot x$

Step 1.3.6

Combine terms.

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Step 1.3.6.1

Multiply $21$ by $5$ .

$105 x^{2} x^{2} - 3 (21 x^{2}) + 5 x^{2} \cdot 1 - 3 \cdot 1 + 10 (7 x^{3}) x + 10 x \cdot x$

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 $x^{2}$ .

$105 (x^{2} x^{2}) - 3 (21 x^{2}) + 5 x^{2} \cdot 1 - 3 \cdot 1 + 10 (7 x^{3}) x + 10 x \cdot x$

Step 1.3.6.2.2

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

$105 x^{2 + 2} - 3 (21 x^{2}) + 5 x^{2} \cdot 1 - 3 \cdot 1 + 10 (7 x^{3}) x + 10 x \cdot x$

Step 1.3.6.2.3

Add $2$ and $2$ .

$105 x^{4} - 3 (21 x^{2}) + 5 x^{2} \cdot 1 - 3 \cdot 1 + 10 (7 x^{3}) x + 10 x \cdot x$

$105 x^{4} - 3 (21 x^{2}) + 5 x^{2} \cdot 1 - 3 \cdot 1 + 10 (7 x^{3}) x + 10 x \cdot x$

Step 1.3.6.3

Multiply $21$ by $- 3$ .

$105 x^{4} - 63 x^{2} + 5 x^{2} \cdot 1 - 3 \cdot 1 + 10 (7 x^{3}) x + 10 x \cdot x$

Step 1.3.6.4

Multiply $5$ by $1$ .

$105 x^{4} - 63 x^{2} + 5 x^{2} - 3 \cdot 1 + 10 (7 x^{3}) x + 10 x \cdot x$

Step 1.3.6.5

Multiply $- 3$ by $1$ .

$105 x^{4} - 63 x^{2} + 5 x^{2} - 3 + 10 (7 x^{3}) x + 10 x \cdot x$

Step 1.3.6.6

Add $- 63 x^{2}$ and $5 x^{2}$ .

$105 x^{4} - 58 x^{2} - 3 + 10 (7 x^{3}) x + 10 x \cdot x$

Step 1.3.6.7

Multiply $7$ by $10$ .

$105 x^{4} - 58 x^{2} - 3 + 70 x^{3} x + 10 x \cdot x$

Step 1.3.6.8

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

$105 x^{4} - 58 x^{2} - 3 + 70 (x^{1} x^{3}) + 10 x \cdot x$

Step 1.3.6.9

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

$105 x^{4} - 58 x^{2} - 3 + 70 x^{1 + 3} + 10 x \cdot x$

Step 1.3.6.10

Add $1$ and $3$ .

$105 x^{4} - 58 x^{2} - 3 + 70 x^{4} + 10 x \cdot x$

Step 1.3.6.11

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

$105 x^{4} - 58 x^{2} - 3 + 70 x^{4} + 10 (x^{1} x)$

Step 1.3.6.12

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

$105 x^{4} - 58 x^{2} - 3 + 70 x^{4} + 10 (x^{1} x^{1})$

Step 1.3.6.13

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

$105 x^{4} - 58 x^{2} - 3 + 70 x^{4} + 10 x^{1 + 1}$

Step 1.3.6.14

Add $1$ and $1$ .

$105 x^{4} - 58 x^{2} - 3 + 70 x^{4} + 10 x^{2}$

Step 1.3.6.15

Add $105 x^{4}$ and $70 x^{4}$ .

$175 x^{4} - 58 x^{2} - 3 + 10 x^{2}$

Step 1.3.6.16

Add $- 58 x^{2}$ and $10 x^{2}$ .

$f' (x) = 175 x^{4} - 48 x^{2} - 3$

$f' (x) = 175 x^{4} - 48 x^{2} - 3$

$f' (x) = 175 x^{4} - 48 x^{2} - 3$

$f' (x) = 175 x^{4} - 48 x^{2} - 3$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $175 x^{4} - 48 x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [175 x^{4}] + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [175 x^{4}] + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [- 3]$

Step 2.2

Evaluate $\frac{d}{d x} [175 x^{4}]$ .

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Step 2.2.1

Since $175$ is constant with respect to $x$ , the derivative of $175 x^{4}$ with respect to $x$ is $175 \frac{d}{d x} [x^{4}]$ .

$175 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [- 3]$

Step 2.2.2

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

$175 (4 x^{3}) + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [- 3]$

Step 2.2.3

Multiply $4$ by $175$ .

$700 x^{3} + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [- 3]$

$700 x^{3} + \frac{d}{d x} [- 48 x^{2}] + \frac{d}{d x} [- 3]$

Step 2.3

Evaluate $\frac{d}{d x} [- 48 x^{2}]$ .

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Step 2.3.1

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48 x^{2}$ with respect to $x$ is $- 48 \frac{d}{d x} [x^{2}]$ .

$700 x^{3} - 48 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]$

Step 2.3.2

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

$700 x^{3} - 48 (2 x) + \frac{d}{d x} [- 3]$

Step 2.3.3

Multiply $2$ by $- 48$ .

$700 x^{3} - 96 x + \frac{d}{d x} [- 3]$

$700 x^{3} - 96 x + \frac{d}{d x} [- 3]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$700 x^{3} - 96 x + 0$

Step 2.4.2

Add $700 x^{3} - 96 x$ and $0$ .

$f'' (x) = 700 x^{3} - 96 x$

$f'' (x) = 700 x^{3} - 96 x$

$f'' (x) = 700 x^{3} - 96 x$

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