Calculus Examples

$y (x) = (9 x^{2} - 7 x) (18 x - \frac{97}{x})$

Step 1

Find the first derivative.

Tap for more steps...

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 $f (x) = 9 x^{2} - 7 x$ and $g (x) = 18 x - \frac{97}{x}$ .

$(9 x^{2} - 7 x) \frac{d}{d x} [18 x - \frac{97}{x}] + (18 x - \frac{97}{x}) \frac{d}{d x} [9 x^{2} - 7 x]$

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

By the Sum Rule, the derivative of $18 x - \frac{97}{x}$ with respect to $x$ is $\frac{d}{d x} [18 x] + \frac{d}{d x} [- \frac{97}{x}]$ .

$(9 x^{2} - 7 x) (\frac{d}{d x} [18 x] + \frac{d}{d x} [- \frac{97}{x}]) + (18 x - \frac{97}{x}) \frac{d}{d x} [9 x^{2} - 7 x]$

Step 1.2.2

Since $18$ is constant with respect to $x$ , the derivative of $18 x$ with respect to $x$ is $18 \frac{d}{d x} [x]$ .

$(9 x^{2} - 7 x) (18 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{97}{x}]) + (18 x - \frac{97}{x}) \frac{d}{d x} [9 x^{2} - 7 x]$

Step 1.2.3

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

$(9 x^{2} - 7 x) (18 \cdot 1 + \frac{d}{d x} [- \frac{97}{x}]) + (18 x - \frac{97}{x}) \frac{d}{d x} [9 x^{2} - 7 x]$

Step 1.2.4

Multiply $18$ by $1$ .

$(9 x^{2} - 7 x) (18 + \frac{d}{d x} [- \frac{97}{x}]) + (18 x - \frac{97}{x}) \frac{d}{d x} [9 x^{2} - 7 x]$

Step 1.2.5

Since $- 97$ is constant with respect to $x$ , the derivative of $- \frac{97}{x}$ with respect to $x$ is $- 97 \frac{d}{d x} [\frac{1}{x}]$ .

$(9 x^{2} - 7 x) (18 - 97 \frac{d}{d x} [\frac{1}{x}]) + (18 x - \frac{97}{x}) \frac{d}{d x} [9 x^{2} - 7 x]$

Step 1.2.6

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$(9 x^{2} - 7 x) (18 - 97 \frac{d}{d x} [x^{- 1}]) + (18 x - \frac{97}{x}) \frac{d}{d x} [9 x^{2} - 7 x]$

Step 1.2.7

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

$(9 x^{2} - 7 x) (18 - 97 (- x^{- 2})) + (18 x - \frac{97}{x}) \frac{d}{d x} [9 x^{2} - 7 x]$

Step 1.2.8

Multiply $- 1$ by $- 97$ .

$(9 x^{2} - 7 x) (18 + 97 x^{- 2}) + (18 x - \frac{97}{x}) \frac{d}{d x} [9 x^{2} - 7 x]$

Step 1.2.9

By the Sum Rule, the derivative of $9 x^{2} - 7 x$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 7 x]$ .

$(9 x^{2} - 7 x) (18 + 97 x^{- 2}) + (18 x - \frac{97}{x}) (\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 7 x])$

Step 1.2.10

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$(9 x^{2} - 7 x) (18 + 97 x^{- 2}) + (18 x - \frac{97}{x}) (9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x])$

Step 1.2.11

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

$(9 x^{2} - 7 x) (18 + 97 x^{- 2}) + (18 x - \frac{97}{x}) (9 (2 x) + \frac{d}{d x} [- 7 x])$

Step 1.2.12

Multiply $2$ by $9$ .

$(9 x^{2} - 7 x) (18 + 97 x^{- 2}) + (18 x - \frac{97}{x}) (18 x + \frac{d}{d x} [- 7 x])$

Step 1.2.13

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

$(9 x^{2} - 7 x) (18 + 97 x^{- 2}) + (18 x - \frac{97}{x}) (18 x - 7 \frac{d}{d x} [x])$

Step 1.2.14

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

$(9 x^{2} - 7 x) (18 + 97 x^{- 2}) + (18 x - \frac{97}{x}) (18 x - 7 \cdot 1)$

Step 1.2.15

Multiply $- 7$ by $1$ .

$(9 x^{2} - 7 x) (18 + 97 x^{- 2}) + (18 x - \frac{97}{x}) (18 x - 7)$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(9 x^{2} - 7 x) (18 + 97 \frac{1}{x^{2}}) + (18 x - \frac{97}{x}) (18 x - 7)$

Step 1.3.2

Combine $97$ and $\frac{1}{x^{2}}$ .

$(9 x^{2} - 7 x) (18 + \frac{97}{x^{2}}) + (18 x - \frac{97}{x}) (18 x - 7)$

Step 1.3.3

Reorder terms.

$(18 + \frac{97}{x^{2}}) (9 x^{2} - 7 x) + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4

Simplify each term.

Tap for more steps...

Step 1.3.4.1

Expand $(18 + \frac{97}{x^{2}}) (9 x^{2} - 7 x)$ using the FOIL Method.

Tap for more steps...

Step 1.3.4.1.1

Apply the distributive property.

$18 (9 x^{2} - 7 x) + \frac{97}{x^{2}} (9 x^{2} - 7 x) + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.1.2

Apply the distributive property.

$18 (9 x^{2}) + 18 (- 7 x) + \frac{97}{x^{2}} (9 x^{2} - 7 x) + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.1.3

Apply the distributive property.

$18 (9 x^{2}) + 18 (- 7 x) + \frac{97}{x^{2}} (9 x^{2}) + \frac{97}{x^{2}} (- 7 x) + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2

Simplify each term.

Tap for more steps...

Step 1.3.4.2.1

Multiply $9$ by $18$ .

$162 x^{2} + 18 (- 7 x) + \frac{97}{x^{2}} (9 x^{2}) + \frac{97}{x^{2}} (- 7 x) + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.2

Multiply $- 7$ by $18$ .

$162 x^{2} - 126 x + \frac{97}{x^{2}} (9 x^{2}) + \frac{97}{x^{2}} (- 7 x) + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.3

Rewrite using the commutative property of multiplication.

$162 x^{2} - 126 x + 9 \frac{97}{x^{2}} x^{2} + \frac{97}{x^{2}} (- 7 x) + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.4

Multiply $9 \frac{97}{x^{2}}$ .

Tap for more steps...

Step 1.3.4.2.4.1

Combine $9$ and $\frac{97}{x^{2}}$ .

$162 x^{2} - 126 x + \frac{9 \cdot 97}{x^{2}} x^{2} + \frac{97}{x^{2}} (- 7 x) + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.4.2

Multiply $9$ by $97$ .

$162 x^{2} - 126 x + \frac{873}{x^{2}} x^{2} + \frac{97}{x^{2}} (- 7 x) + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.5

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 1.3.4.2.5.1

Cancel the common factor.

$162 x^{2} - 126 x + \frac{873}{x^{2}} x^{2} + \frac{97}{x^{2}} (- 7 x) + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.5.2

Rewrite the expression.

$162 x^{2} - 126 x + 873 + \frac{97}{x^{2}} (- 7 x) + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.6

Rewrite using the commutative property of multiplication.

$162 x^{2} - 126 x + 873 - 7 \frac{97}{x^{2}} x + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.7

Multiply $- 7 \frac{97}{x^{2}}$ .

Tap for more steps...

Step 1.3.4.2.7.1

Combine $- 7$ and $\frac{97}{x^{2}}$ .

$162 x^{2} - 126 x + 873 + \frac{- 7 \cdot 97}{x^{2}} x + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.7.2

Multiply $- 7$ by $97$ .

$162 x^{2} - 126 x + 873 + \frac{- 679}{x^{2}} x + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.8

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.3.4.2.8.1

Factor $x$ out of $x^{2}$ .

$162 x^{2} - 126 x + 873 + \frac{- 679}{x \cdot x} x + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.8.2

Cancel the common factor.

$162 x^{2} - 126 x + 873 + \frac{- 679}{x \cdot x} x + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.8.3

Rewrite the expression.

$162 x^{2} - 126 x + 873 + \frac{- 679}{x} + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.2.9

Move the negative in front of the fraction.

$162 x^{2} - 126 x + 873 - \frac{679}{x} + (18 x - 7) (18 x - \frac{97}{x})$

Step 1.3.4.3

Expand $(18 x - 7) (18 x - \frac{97}{x})$ using the FOIL Method.

Tap for more steps...

Step 1.3.4.3.1

Apply the distributive property.

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 18 x (18 x - \frac{97}{x}) - 7 (18 x - \frac{97}{x})$

Step 1.3.4.3.2

Apply the distributive property.

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 18 x (18 x) + 18 x (- \frac{97}{x}) - 7 (18 x - \frac{97}{x})$

Step 1.3.4.3.3

Apply the distributive property.

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 18 x (18 x) + 18 x (- \frac{97}{x}) - 7 (18 x) - 7 (- \frac{97}{x})$

Step 1.3.4.4

Simplify each term.

Tap for more steps...

Step 1.3.4.4.1

Rewrite using the commutative property of multiplication.

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 18 \cdot 18 x \cdot x + 18 x (- \frac{97}{x}) - 7 (18 x) - 7 (- \frac{97}{x})$

Step 1.3.4.4.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.3.4.4.2.1

Move $x$ .

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 18 \cdot 18 (x \cdot x) + 18 x (- \frac{97}{x}) - 7 (18 x) - 7 (- \frac{97}{x})$

Step 1.3.4.4.2.2

Multiply $x$ by $x$ .

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 18 \cdot 18 x^{2} + 18 x (- \frac{97}{x}) - 7 (18 x) - 7 (- \frac{97}{x})$

Step 1.3.4.4.3

Multiply $18$ by $18$ .

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 324 x^{2} + 18 x (- \frac{97}{x}) - 7 (18 x) - 7 (- \frac{97}{x})$

Step 1.3.4.4.4

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.3.4.4.4.1

Move the leading negative in $- \frac{97}{x}$ into the numerator.

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 324 x^{2} + 18 x \frac{- 97}{x} - 7 (18 x) - 7 (- \frac{97}{x})$

Step 1.3.4.4.4.2

Factor $x$ out of $18 x$ .

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 324 x^{2} + x \cdot 18 \frac{- 97}{x} - 7 (18 x) - 7 (- \frac{97}{x})$

Step 1.3.4.4.4.3

Cancel the common factor.

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 324 x^{2} + x \cdot 18 \frac{- 97}{x} - 7 (18 x) - 7 (- \frac{97}{x})$

Step 1.3.4.4.4.4

Rewrite the expression.

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 324 x^{2} + 18 \cdot - 97 - 7 (18 x) - 7 (- \frac{97}{x})$

Step 1.3.4.4.5

Multiply $18$ by $- 97$ .

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 324 x^{2} - 1746 - 7 (18 x) - 7 (- \frac{97}{x})$

Step 1.3.4.4.6

Multiply $18$ by $- 7$ .

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 324 x^{2} - 1746 - 126 x - 7 (- \frac{97}{x})$

Step 1.3.4.4.7

Multiply $- 7 (- \frac{97}{x})$ .

Tap for more steps...

Step 1.3.4.4.7.1

Multiply $- 1$ by $- 7$ .

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 324 x^{2} - 1746 - 126 x + 7 \frac{97}{x}$

Step 1.3.4.4.7.2

Combine $7$ and $\frac{97}{x}$ .

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 324 x^{2} - 1746 - 126 x + \frac{7 \cdot 97}{x}$

Step 1.3.4.4.7.3

Multiply $7$ by $97$ .

$162 x^{2} - 126 x + 873 - \frac{679}{x} + 324 x^{2} - 1746 - 126 x + \frac{679}{x}$

Step 1.3.5

Combine the opposite terms in $162 x^{2} - 126 x + 873 - \frac{679}{x} + 324 x^{2} - 1746 - 126 x + \frac{679}{x}$ .

Tap for more steps...

Step 1.3.5.1

Add $- \frac{679}{x}$ and $\frac{679}{x}$ .

$162 x^{2} - 126 x + 873 + 324 x^{2} - 1746 - 126 x + 0$

Step 1.3.5.2

Add $162 x^{2} - 126 x + 873 + 324 x^{2} - 1746 - 126 x$ and $0$ .

$162 x^{2} - 126 x + 873 + 324 x^{2} - 1746 - 126 x$

Step 1.3.6

Add $162 x^{2}$ and $324 x^{2}$ .

$486 x^{2} - 126 x + 873 - 1746 - 126 x$

Step 1.3.7

Subtract $126 x$ from $- 126 x$ .

$486 x^{2} - 252 x + 873 - 1746$

Step 1.3.8

Subtract $1746$ from $873$ .

$f' (x) = 486 x^{2} - 252 x - 873$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $486 x^{2} - 252 x - 873$ with respect to $x$ is $\frac{d}{d x} [486 x^{2}] + \frac{d}{d x} [- 252 x] + \frac{d}{d x} [- 873]$ .

$\frac{d}{d x} [486 x^{2}] + \frac{d}{d x} [- 252 x] + \frac{d}{d x} [- 873]$

Step 2.2

Evaluate $\frac{d}{d x} [486 x^{2}]$ .

Tap for more steps...

Step 2.2.1

Since $486$ is constant with respect to $x$ , the derivative of $486 x^{2}$ with respect to $x$ is $486 \frac{d}{d x} [x^{2}]$ .

$486 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 252 x] + \frac{d}{d x} [- 873]$

Step 2.2.2

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

$486 (2 x) + \frac{d}{d x} [- 252 x] + \frac{d}{d x} [- 873]$

Step 2.2.3

Multiply $2$ by $486$ .

$972 x + \frac{d}{d x} [- 252 x] + \frac{d}{d x} [- 873]$

Step 2.3

Evaluate $\frac{d}{d x} [- 252 x]$ .

Tap for more steps...

Step 2.3.1

Since $- 252$ is constant with respect to $x$ , the derivative of $- 252 x$ with respect to $x$ is $- 252 \frac{d}{d x} [x]$ .

$972 x - 252 \frac{d}{d x} [x] + \frac{d}{d x} [- 873]$

Step 2.3.2

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

$972 x - 252 \cdot 1 + \frac{d}{d x} [- 873]$

Step 2.3.3

Multiply $- 252$ by $1$ .

$972 x - 252 + \frac{d}{d x} [- 873]$

Step 2.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.4.1

Since $- 873$ is constant with respect to $x$ , the derivative of $- 873$ with respect to $x$ is $0$ .

$972 x - 252 + 0$

Step 2.4.2

Add $972 x - 252$ and $0$ .

$f'' (x) = 972 x - 252$

Step 3

The second derivative of $y (x)$ with respect to $x$ is $972 x - 252$ .

$972 x - 252$