Calculus Examples

$y = \frac{12 e^{6 x} - 9 e^{8 x}}{3 e^{3 x}}$

Step 1

Cancel the common factor of $12 e^{6 x} - 9 e^{8 x}$ and $3$ .

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Step 1.1

Factor $3$ out of $12 e^{6 x}$ .

$\frac{d}{d x} [\frac{3 (4 e^{6 x}) - 9 e^{8 x}}{3 e^{3 x}}]$

Step 1.2

Factor $3$ out of $- 9 e^{8 x}$ .

$\frac{d}{d x} [\frac{3 (4 e^{6 x}) + 3 (- 3 e^{8 x})}{3 e^{3 x}}]$

Step 1.3

Factor $3$ out of $3 (4 e^{6 x}) + 3 (- 3 e^{8 x})$ .

$\frac{d}{d x} [\frac{3 (4 e^{6 x} - 3 e^{8 x})}{3 e^{3 x}}]$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $3$ out of $3 e^{3 x}$ .

$\frac{d}{d x} [\frac{3 (4 e^{6 x} - 3 e^{8 x})}{3 (e^{3 x})}]$

Step 1.4.2

Cancel the common factor.

$\frac{d}{d x} [\frac{3 (4 e^{6 x} - 3 e^{8 x})}{3 e^{3 x}}]$

Step 1.4.3

Rewrite the expression.

$\frac{d}{d x} [\frac{4 e^{6 x} - 3 e^{8 x}}{e^{3 x}}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 4 e^{6 x} - 3 e^{8 x}$ and $g (x) = e^{3 x}$ .

$\frac{e^{3 x} \frac{d}{d x} [4 e^{6 x} - 3 e^{8 x}] - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{{(e^{3 x})}^{2}}$

Step 3

Differentiate.

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Step 3.1

Multiply the exponents in ${(e^{3 x})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{e^{3 x} \frac{d}{d x} [4 e^{6 x} - 3 e^{8 x}] - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{3 x \cdot 2}}$

Step 3.1.2

Multiply $2$ by $3$ .

$\frac{e^{3 x} \frac{d}{d x} [4 e^{6 x} - 3 e^{8 x}] - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 3.2

By the Sum Rule, the derivative of $4 e^{6 x} - 3 e^{8 x}$ with respect to $x$ is $\frac{d}{d x} [4 e^{6 x}] + \frac{d}{d x} [- 3 e^{8 x}]$ .

$\frac{e^{3 x} (\frac{d}{d x} [4 e^{6 x}] + \frac{d}{d x} [- 3 e^{8 x}]) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 3.3

Since $4$ is constant with respect to $x$ , the derivative of $4 e^{6 x}$ with respect to $x$ is $4 \frac{d}{d x} [e^{6 x}]$ .

$\frac{e^{3 x} (4 \frac{d}{d x} [e^{6 x}] + \frac{d}{d x} [- 3 e^{8 x}]) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 6 x$ .

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Step 4.1

To apply the Chain Rule, set $u_{1}$ as $6 x$ .

$\frac{e^{3 x} (4 (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [6 x]) + \frac{d}{d x} [- 3 e^{8 x}]) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$\frac{e^{3 x} (4 (e^{u_{1}} \frac{d}{d x} [6 x]) + \frac{d}{d x} [- 3 e^{8 x}]) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 4.3

Replace all occurrences of $u_{1}$ with $6 x$ .

$\frac{e^{3 x} (4 (e^{6 x} \frac{d}{d x} [6 x]) + \frac{d}{d x} [- 3 e^{8 x}]) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 5

Differentiate.

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Step 5.1

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

$\frac{e^{3 x} (4 e^{6 x} (6 \frac{d}{d x} [x]) + \frac{d}{d x} [- 3 e^{8 x}]) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 5.2

Multiply $6$ by $4$ .

$\frac{e^{3 x} (24 e^{6 x} \frac{d}{d x} [x] + \frac{d}{d x} [- 3 e^{8 x}]) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 5.3

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

$\frac{e^{3 x} (24 e^{6 x} \cdot 1 + \frac{d}{d x} [- 3 e^{8 x}]) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 5.4

Multiply $24$ by $1$ .

$\frac{e^{3 x} (24 e^{6 x} + \frac{d}{d x} [- 3 e^{8 x}]) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 5.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 e^{8 x}$ with respect to $x$ is $- 3 \frac{d}{d x} [e^{8 x}]$ .

$\frac{e^{3 x} (24 e^{6 x} - 3 \frac{d}{d x} [e^{8 x}]) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 8 x$ .

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Step 6.1

To apply the Chain Rule, set $u_{2}$ as $8 x$ .

$\frac{e^{3 x} (24 e^{6 x} - 3 (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [8 x])) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 6.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$\frac{e^{3 x} (24 e^{6 x} - 3 (e^{u_{2}} \frac{d}{d x} [8 x])) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 6.3

Replace all occurrences of $u_{2}$ with $8 x$ .

$\frac{e^{3 x} (24 e^{6 x} - 3 (e^{8 x} \frac{d}{d x} [8 x])) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 7

Differentiate.

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Step 7.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$\frac{e^{3 x} (24 e^{6 x} - 3 e^{8 x} (8 \frac{d}{d x} [x])) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 7.2

Multiply $8$ by $- 3$ .

$\frac{e^{3 x} (24 e^{6 x} - 24 e^{8 x} \frac{d}{d x} [x]) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 7.3

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

$\frac{e^{3 x} (24 e^{6 x} - 24 e^{8 x} \cdot 1) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 7.4

Multiply $- 24$ by $1$ .

$\frac{e^{3 x} (24 e^{6 x} - 24 e^{8 x}) - (4 e^{6 x} - 3 e^{8 x}) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 3 x$ .

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Step 8.1

To apply the Chain Rule, set $u_{3}$ as $3 x$ .

$\frac{e^{3 x} (24 e^{6 x} - 24 e^{8 x}) - (4 e^{6 x} - 3 e^{8 x}) (\frac{d}{d u_{3}} [e^{u_{3}}] \frac{d}{d x} [3 x])}{e^{6 x}}$

Step 8.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

$\frac{e^{3 x} (24 e^{6 x} - 24 e^{8 x}) - (4 e^{6 x} - 3 e^{8 x}) (e^{u_{3}} \frac{d}{d x} [3 x])}{e^{6 x}}$

Step 8.3

Replace all occurrences of $u_{3}$ with $3 x$ .

$\frac{e^{3 x} (24 e^{6 x} - 24 e^{8 x}) - (4 e^{6 x} - 3 e^{8 x}) (e^{3 x} \frac{d}{d x} [3 x])}{e^{6 x}}$

Step 9

Differentiate.

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Step 9.1

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

$\frac{e^{3 x} (24 e^{6 x} - 24 e^{8 x}) - (4 e^{6 x} - 3 e^{8 x}) (e^{3 x} (3 \frac{d}{d x} [x]))}{e^{6 x}}$

Step 9.2

Multiply $3$ by $- 1$ .

$\frac{e^{3 x} (24 e^{6 x} - 24 e^{8 x}) - 3 (4 e^{6 x} - 3 e^{8 x}) (e^{3 x} (\frac{d}{d x} [x]))}{e^{6 x}}$

Step 9.3

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

$\frac{e^{3 x} (24 e^{6 x} - 24 e^{8 x}) - 3 (4 e^{6 x} - 3 e^{8 x}) (e^{3 x} \cdot 1)}{e^{6 x}}$

Step 9.4

Multiply $e^{3 x}$ by $1$ .

$\frac{e^{3 x} (24 e^{6 x} - 24 e^{8 x}) - 3 (4 e^{6 x} - 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{e^{3 x} (24 e^{6 x}) + e^{3 x} (- 24 e^{8 x}) - 3 (4 e^{6 x} - 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.2

Apply the distributive property.

$\frac{e^{3 x} (24 e^{6 x}) + e^{3 x} (- 24 e^{8 x}) + (- 3 (4 e^{6 x}) - 3 (- 3 e^{8 x})) e^{3 x}}{e^{6 x}}$

Step 10.3

Apply the distributive property.

$\frac{e^{3 x} (24 e^{6 x}) + e^{3 x} (- 24 e^{8 x}) - 3 (4 e^{6 x}) e^{3 x} - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4

Simplify the numerator.

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Step 10.4.1

Simplify each term.

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Step 10.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{24 e^{3 x} e^{6 x} + e^{3 x} (- 24 e^{8 x}) - 3 (4 e^{6 x}) e^{3 x} - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.2

Multiply $e^{3 x}$ by $e^{6 x}$ by adding the exponents.

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Step 10.4.1.2.1

Move $e^{6 x}$ .

$\frac{24 (e^{6 x} e^{3 x}) + e^{3 x} (- 24 e^{8 x}) - 3 (4 e^{6 x}) e^{3 x} - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.2.2

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

$\frac{24 e^{6 x + 3 x} + e^{3 x} (- 24 e^{8 x}) - 3 (4 e^{6 x}) e^{3 x} - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.2.3

Add $6 x$ and $3 x$ .

$\frac{24 e^{9 x} + e^{3 x} (- 24 e^{8 x}) - 3 (4 e^{6 x}) e^{3 x} - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.3

Rewrite using the commutative property of multiplication.

$\frac{24 e^{9 x} - 24 e^{3 x} e^{8 x} - 3 (4 e^{6 x}) e^{3 x} - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.4

Multiply $e^{3 x}$ by $e^{8 x}$ by adding the exponents.

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Step 10.4.1.4.1

Move $e^{8 x}$ .

$\frac{24 e^{9 x} - 24 (e^{8 x} e^{3 x}) - 3 (4 e^{6 x}) e^{3 x} - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.4.2

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

$\frac{24 e^{9 x} - 24 e^{8 x + 3 x} - 3 (4 e^{6 x}) e^{3 x} - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.4.3

Add $8 x$ and $3 x$ .

$\frac{24 e^{9 x} - 24 e^{11 x} - 3 (4 e^{6 x}) e^{3 x} - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.5

Multiply $e^{6 x}$ by $e^{3 x}$ by adding the exponents.

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Step 10.4.1.5.1

Move $e^{3 x}$ .

$\frac{24 e^{9 x} - 24 e^{11 x} - 3 (4 (e^{3 x} e^{6 x})) - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.5.2

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

$\frac{24 e^{9 x} - 24 e^{11 x} - 3 (4 e^{3 x + 6 x}) - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.5.3

Add $3 x$ and $6 x$ .

$\frac{24 e^{9 x} - 24 e^{11 x} - 3 (4 e^{9 x}) - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.6

Multiply $4$ by $- 3$ .

$\frac{24 e^{9 x} - 24 e^{11 x} - 12 e^{9 x} - 3 (- 3 e^{8 x}) e^{3 x}}{e^{6 x}}$

Step 10.4.1.7

Multiply $e^{8 x}$ by $e^{3 x}$ by adding the exponents.

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Step 10.4.1.7.1

Move $e^{3 x}$ .

$\frac{24 e^{9 x} - 24 e^{11 x} - 12 e^{9 x} - 3 (- 3 (e^{3 x} e^{8 x}))}{e^{6 x}}$

Step 10.4.1.7.2

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

$\frac{24 e^{9 x} - 24 e^{11 x} - 12 e^{9 x} - 3 (- 3 e^{3 x + 8 x})}{e^{6 x}}$

Step 10.4.1.7.3

Add $3 x$ and $8 x$ .

$\frac{24 e^{9 x} - 24 e^{11 x} - 12 e^{9 x} - 3 (- 3 e^{11 x})}{e^{6 x}}$

Step 10.4.1.8

Multiply $- 3$ by $- 3$ .

$\frac{24 e^{9 x} - 24 e^{11 x} - 12 e^{9 x} + 9 e^{11 x}}{e^{6 x}}$

Step 10.4.2

Subtract $12 e^{9 x}$ from $24 e^{9 x}$ .

$\frac{12 e^{9 x} - 24 e^{11 x} + 9 e^{11 x}}{e^{6 x}}$

Step 10.4.3

Add $- 24 e^{11 x}$ and $9 e^{11 x}$ .

$\frac{12 e^{9 x} - 15 e^{11 x}}{e^{6 x}}$

Step 10.5

Factor $3$ out of $12 e^{9 x} - 15 e^{11 x}$ .

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Step 10.5.1

Factor $3$ out of $12 e^{9 x}$ .

$\frac{3 (4 e^{9 x}) - 15 e^{11 x}}{e^{6 x}}$

Step 10.5.2

Factor $3$ out of $- 15 e^{11 x}$ .

$\frac{3 (4 e^{9 x}) + 3 (- 5 e^{11 x})}{e^{6 x}}$

Step 10.5.3

Factor $3$ out of $3 (4 e^{9 x}) + 3 (- 5 e^{11 x})$ .

$\frac{3 (4 e^{9 x} - 5 e^{11 x})}{e^{6 x}}$