Calculus Examples

$g (x) = (3 x^{5} - 7 e^{x}) (9 x^{4} - 6 x^{- 1})$

Step 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 $f (x) = 3 x^{5} - 7 e^{x}$ and $g (x) = 9 x^{4} - 6 x^{- 1}$ .

$(3 x^{5} - 7 e^{x}) \frac{d}{d x} [9 x^{4} - 6 x^{- 1}] + (9 x^{4} - 6 x^{- 1}) \frac{d}{d x} [3 x^{5} - 7 e^{x}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $9 x^{4} - 6 x^{- 1}$ with respect to $x$ is $\frac{d}{d x} [9 x^{4}] + \frac{d}{d x} [- 6 x^{- 1}]$ .

$(3 x^{5} - 7 e^{x}) (\frac{d}{d x} [9 x^{4}] + \frac{d}{d x} [- 6 x^{- 1}]) + (9 x^{4} - 6 x^{- 1}) \frac{d}{d x} [3 x^{5} - 7 e^{x}]$

Step 2.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{4}$ with respect to $x$ is $9 \frac{d}{d x} [x^{4}]$ .

$(3 x^{5} - 7 e^{x}) (9 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 6 x^{- 1}]) + (9 x^{4} - 6 x^{- 1}) \frac{d}{d x} [3 x^{5} - 7 e^{x}]$

Step 2.3

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

$(3 x^{5} - 7 e^{x}) (9 (4 x^{3}) + \frac{d}{d x} [- 6 x^{- 1}]) + (9 x^{4} - 6 x^{- 1}) \frac{d}{d x} [3 x^{5} - 7 e^{x}]$

Step 2.4

Multiply $4$ by $9$ .

$(3 x^{5} - 7 e^{x}) (36 x^{3} + \frac{d}{d x} [- 6 x^{- 1}]) + (9 x^{4} - 6 x^{- 1}) \frac{d}{d x} [3 x^{5} - 7 e^{x}]$

Step 2.5

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{- 1}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{- 1}]$ .

$(3 x^{5} - 7 e^{x}) (36 x^{3} - 6 \frac{d}{d x} [x^{- 1}]) + (9 x^{4} - 6 x^{- 1}) \frac{d}{d x} [3 x^{5} - 7 e^{x}]$

Step 2.6

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

$(3 x^{5} - 7 e^{x}) (36 x^{3} - 6 (- x^{- 2})) + (9 x^{4} - 6 x^{- 1}) \frac{d}{d x} [3 x^{5} - 7 e^{x}]$

Step 2.7

Multiply $- 1$ by $- 6$ .

$(3 x^{5} - 7 e^{x}) (36 x^{3} + 6 x^{- 2}) + (9 x^{4} - 6 x^{- 1}) \frac{d}{d x} [3 x^{5} - 7 e^{x}]$

Step 2.8

By the Sum Rule, the derivative of $3 x^{5} - 7 e^{x}$ with respect to $x$ is $\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [- 7 e^{x}]$ .

$(3 x^{5} - 7 e^{x}) (36 x^{3} + 6 x^{- 2}) + (9 x^{4} - 6 x^{- 1}) (\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [- 7 e^{x}])$

Step 2.9

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{5}$ with respect to $x$ is $3 \frac{d}{d x} [x^{5}]$ .

$(3 x^{5} - 7 e^{x}) (36 x^{3} + 6 x^{- 2}) + (9 x^{4} - 6 x^{- 1}) (3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 7 e^{x}])$

Step 2.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$(3 x^{5} - 7 e^{x}) (36 x^{3} + 6 x^{- 2}) + (9 x^{4} - 6 x^{- 1}) (3 (5 x^{4}) + \frac{d}{d x} [- 7 e^{x}])$

Step 2.11

Multiply $5$ by $3$ .

$(3 x^{5} - 7 e^{x}) (36 x^{3} + 6 x^{- 2}) + (9 x^{4} - 6 x^{- 1}) (15 x^{4} + \frac{d}{d x} [- 7 e^{x}])$

Step 2.12

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

$(3 x^{5} - 7 e^{x}) (36 x^{3} + 6 x^{- 2}) + (9 x^{4} - 6 x^{- 1}) (15 x^{4} - 7 \frac{d}{d x} [e^{x}])$

Step 3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$(3 x^{5} - 7 e^{x}) (36 x^{3} + 6 x^{- 2}) + (9 x^{4} - 6 x^{- 1}) (15 x^{4} - 7 e^{x})$

Step 4

Simplify.

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Step 4.1

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

$(3 x^{5} - 7 e^{x}) (36 x^{3} + 6 \frac{1}{x^{2}}) + (9 x^{4} - 6 x^{- 1}) (15 x^{4} - 7 e^{x})$

Step 4.2

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

$(3 x^{5} - 7 e^{x}) (36 x^{3} + 6 \frac{1}{x^{2}}) + (9 x^{4} - 6 \frac{1}{x}) (15 x^{4} - 7 e^{x})$

Step 4.3

Combine terms.

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Step 4.3.1

Combine $6$ and $\frac{1}{x^{2}}$ .

$(3 x^{5} - 7 e^{x}) (36 x^{3} + \frac{6}{x^{2}}) + (9 x^{4} - 6 \frac{1}{x}) (15 x^{4} - 7 e^{x})$

Step 4.3.2

Combine $- 6$ and $\frac{1}{x}$ .

$(3 x^{5} - 7 e^{x}) (36 x^{3} + \frac{6}{x^{2}}) + (9 x^{4} + \frac{- 6}{x}) (15 x^{4} - 7 e^{x})$

Step 4.3.3

Move the negative in front of the fraction.

$(3 x^{5} - 7 e^{x}) (36 x^{3} + \frac{6}{x^{2}}) + (9 x^{4} - \frac{6}{x}) (15 x^{4} - 7 e^{x})$

Step 4.4

Reorder terms.

$(36 x^{3} + \frac{6}{x^{2}}) (3 x^{5} - 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5

Simplify each term.

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Step 4.5.1

Expand $(36 x^{3} + \frac{6}{x^{2}}) (3 x^{5} - 7 e^{x})$ using the FOIL Method.

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Step 4.5.1.1

Apply the distributive property.

$36 x^{3} (3 x^{5} - 7 e^{x}) + \frac{6}{x^{2}} (3 x^{5} - 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.1.2

Apply the distributive property.

$36 x^{3} (3 x^{5}) + 36 x^{3} (- 7 e^{x}) + \frac{6}{x^{2}} (3 x^{5} - 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.1.3

Apply the distributive property.

$36 x^{3} (3 x^{5}) + 36 x^{3} (- 7 e^{x}) + \frac{6}{x^{2}} (3 x^{5}) + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2

Simplify each term.

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Step 4.5.2.1

Rewrite using the commutative property of multiplication.

$36 \cdot 3 x^{3} x^{5} + 36 x^{3} (- 7 e^{x}) + \frac{6}{x^{2}} (3 x^{5}) + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.2

Multiply $x^{3}$ by $x^{5}$ by adding the exponents.

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Step 4.5.2.2.1

Move $x^{5}$ .

$36 \cdot 3 (x^{5} x^{3}) + 36 x^{3} (- 7 e^{x}) + \frac{6}{x^{2}} (3 x^{5}) + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.2.2

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

$36 \cdot 3 x^{5 + 3} + 36 x^{3} (- 7 e^{x}) + \frac{6}{x^{2}} (3 x^{5}) + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.2.3

Add $5$ and $3$ .

$36 \cdot 3 x^{8} + 36 x^{3} (- 7 e^{x}) + \frac{6}{x^{2}} (3 x^{5}) + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.3

Multiply $36$ by $3$ .

$108 x^{8} + 36 x^{3} (- 7 e^{x}) + \frac{6}{x^{2}} (3 x^{5}) + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.4

Rewrite using the commutative property of multiplication.

$108 x^{8} + 36 \cdot - 7 x^{3} e^{x} + \frac{6}{x^{2}} (3 x^{5}) + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.5

Multiply $36$ by $- 7$ .

$108 x^{8} - 252 x^{3} e^{x} + \frac{6}{x^{2}} (3 x^{5}) + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.6

Rewrite using the commutative property of multiplication.

$108 x^{8} - 252 x^{3} e^{x} + 3 \frac{6}{x^{2}} x^{5} + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.7

Multiply $3 \frac{6}{x^{2}}$ .

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Step 4.5.2.7.1

Combine $3$ and $\frac{6}{x^{2}}$ .

$108 x^{8} - 252 x^{3} e^{x} + \frac{3 \cdot 6}{x^{2}} x^{5} + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.7.2

Multiply $3$ by $6$ .

$108 x^{8} - 252 x^{3} e^{x} + \frac{18}{x^{2}} x^{5} + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.8

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

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Step 4.5.2.8.1

Factor $x^{2}$ out of $x^{5}$ .

$108 x^{8} - 252 x^{3} e^{x} + \frac{18}{x^{2}} (x^{2} x^{3}) + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.8.2

Cancel the common factor.

$108 x^{8} - 252 x^{3} e^{x} + \frac{18}{x^{2}} (x^{2} x^{3}) + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.8.3

Rewrite the expression.

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} + \frac{6}{x^{2}} (- 7 e^{x}) + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.9

Rewrite using the commutative property of multiplication.

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - 7 \frac{6}{x^{2}} e^{x} + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.10

Multiply $- 7 \frac{6}{x^{2}}$ .

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Step 4.5.2.10.1

Combine $- 7$ and $\frac{6}{x^{2}}$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} + \frac{- 7 \cdot 6}{x^{2}} e^{x} + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.10.2

Multiply $- 7$ by $6$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} + \frac{- 42}{x^{2}} e^{x} + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.11

Move the negative in front of the fraction.

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42}{x^{2}} e^{x} + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.12

Combine $e^{x}$ and $\frac{42}{x^{2}}$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{e^{x} \cdot 42}{x^{2}} + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.2.13

Move $42$ to the left of $e^{x}$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + (15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$

Step 4.5.3

Expand $(15 x^{4} - 7 e^{x}) (9 x^{4} - \frac{6}{x})$ using the FOIL Method.

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Step 4.5.3.1

Apply the distributive property.

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 15 x^{4} (9 x^{4} - \frac{6}{x}) - 7 e^{x} (9 x^{4} - \frac{6}{x})$

Step 4.5.3.2

Apply the distributive property.

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 15 x^{4} (9 x^{4}) + 15 x^{4} (- \frac{6}{x}) - 7 e^{x} (9 x^{4} - \frac{6}{x})$

Step 4.5.3.3

Apply the distributive property.

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 15 x^{4} (9 x^{4}) + 15 x^{4} (- \frac{6}{x}) - 7 e^{x} (9 x^{4}) - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4

Simplify each term.

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Step 4.5.4.1

Rewrite using the commutative property of multiplication.

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 15 \cdot 9 x^{4} x^{4} + 15 x^{4} (- \frac{6}{x}) - 7 e^{x} (9 x^{4}) - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.2

Multiply $x^{4}$ by $x^{4}$ by adding the exponents.

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Step 4.5.4.2.1

Move $x^{4}$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 15 \cdot 9 (x^{4} x^{4}) + 15 x^{4} (- \frac{6}{x}) - 7 e^{x} (9 x^{4}) - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.2.2

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

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 15 \cdot 9 x^{4 + 4} + 15 x^{4} (- \frac{6}{x}) - 7 e^{x} (9 x^{4}) - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.2.3

Add $4$ and $4$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 15 \cdot 9 x^{8} + 15 x^{4} (- \frac{6}{x}) - 7 e^{x} (9 x^{4}) - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.3

Multiply $15$ by $9$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} + 15 x^{4} (- \frac{6}{x}) - 7 e^{x} (9 x^{4}) - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.4

Cancel the common factor of $x$ .

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Step 4.5.4.4.1

Move the leading negative in $- \frac{6}{x}$ into the numerator.

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} + 15 x^{4} \frac{- 6}{x} - 7 e^{x} (9 x^{4}) - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.4.2

Factor $x$ out of $15 x^{4}$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} + x (15 x^{3}) \frac{- 6}{x} - 7 e^{x} (9 x^{4}) - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.4.3

Cancel the common factor.

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} + x (15 x^{3}) \frac{- 6}{x} - 7 e^{x} (9 x^{4}) - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.4.4

Rewrite the expression.

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} + 15 x^{3} \cdot - 6 - 7 e^{x} (9 x^{4}) - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.5

Multiply $- 6$ by $15$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} - 90 x^{3} - 7 e^{x} (9 x^{4}) - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.6

Rewrite using the commutative property of multiplication.

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} - 90 x^{3} - 7 \cdot 9 e^{x} x^{4} - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.7

Multiply $- 7$ by $9$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} - 90 x^{3} - 63 e^{x} x^{4} - 7 e^{x} (- \frac{6}{x})$

Step 4.5.4.8

Multiply $- 7 e^{x} (- \frac{6}{x})$ .

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Step 4.5.4.8.1

Multiply $- 1$ by $- 7$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} - 90 x^{3} - 63 e^{x} x^{4} + 7 e^{x} \frac{6}{x}$

Step 4.5.4.8.2

Combine $\frac{6}{x}$ and $7$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} - 90 x^{3} - 63 e^{x} x^{4} + \frac{6 \cdot 7}{x} e^{x}$

Step 4.5.4.8.3

Multiply $6$ by $7$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} - 90 x^{3} - 63 e^{x} x^{4} + \frac{42}{x} e^{x}$

Step 4.5.4.8.4

Combine $\frac{42}{x}$ and $e^{x}$ .

$108 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} + 135 x^{8} - 90 x^{3} - 63 e^{x} x^{4} + \frac{42 e^{x}}{x}$

Step 4.6

Add $108 x^{8}$ and $135 x^{8}$ .

$243 x^{8} - 252 x^{3} e^{x} + 18 x^{3} - \frac{42 e^{x}}{x^{2}} - 90 x^{3} - 63 e^{x} x^{4} + \frac{42 e^{x}}{x}$

Step 4.7

Subtract $90 x^{3}$ from $18 x^{3}$ .

$243 x^{8} - 252 x^{3} e^{x} - 72 x^{3} - \frac{42 e^{x}}{x^{2}} - 63 e^{x} x^{4} + \frac{42 e^{x}}{x}$

Step 4.8

Reorder factors in $243 x^{8} - 252 x^{3} e^{x} - 72 x^{3} - \frac{42 e^{x}}{x^{2}} - 63 e^{x} x^{4} + \frac{42 e^{x}}{x}$ .

$243 x^{8} - 252 x^{3} e^{x} - 72 x^{3} - \frac{42 e^{x}}{x^{2}} - 63 x^{4} e^{x} + \frac{42 e^{x}}{x}$