Calculus Examples

$- \frac{5}{x^{3}} \cdot e^{- 3 x^{4} + 4 x^{2}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $e^{- 3 x^{4} + 4 x^{2}}$ and $\frac{5}{x^{3}}$ .

$\frac{d}{d x} [- \frac{e^{- 3 x^{4} + 4 x^{2}} \cdot 5}{x^{3}}]$

Step 1.2

Move $5$ to the left of $e^{- 3 x^{4} + 4 x^{2}}$ .

$\frac{d}{d x} [- \frac{5 \cdot e^{- 3 x^{4} + 4 x^{2}}}{x^{3}}]$

Step 1.3

Since $- 5$ is constant with respect to $x$ , the derivative of $- \frac{5 e^{- 3 x^{4} + 4 x^{2}}}{x^{3}}$ with respect to $x$ is $- 5 \frac{d}{d x} [\frac{e^{- 3 x^{4} + 4 x^{2}}}{x^{3}}]$ .

$- 5 \frac{d}{d x} [\frac{e^{- 3 x^{4} + 4 x^{2}}}{x^{3}}]$

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 $f (x) = e^{- 3 x^{4} + 4 x^{2}}$ and $g (x) = x^{3}$ .

$- 5 \frac{x^{3} \frac{d}{d x} [e^{- 3 x^{4} + 4 x^{2}}] - e^{- 3 x^{4} + 4 x^{2}} \frac{d}{d x} [x^{3}]}{{(x^{3})}^{2}}$

Step 3

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

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Step 3.1

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

$- 5 \frac{x^{3} \frac{d}{d x} [e^{- 3 x^{4} + 4 x^{2}}] - e^{- 3 x^{4} + 4 x^{2}} \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}}$

Step 3.2

Multiply $3$ by $2$ .

$- 5 \frac{x^{3} \frac{d}{d x} [e^{- 3 x^{4} + 4 x^{2}}] - e^{- 3 x^{4} + 4 x^{2}} \frac{d}{d x} [x^{3}]}{x^{6}}$

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 $g (x) = - 3 x^{4} + 4 x^{2}$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $- 3 x^{4} + 4 x^{2}$ .

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

Step 4.2

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

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

Step 4.3

Replace all occurrences of $u$ with $- 3 x^{4} + 4 x^{2}$ .

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{4}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{4}]$ .

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

Step 5.3

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

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

Step 5.4

Multiply $4$ by $- 3$ .

$- 5 \frac{x^{3} (e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{3} + \frac{d}{d x} [4 x^{2}])) - e^{- 3 x^{4} + 4 x^{2}} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5.5

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$- 5 \frac{x^{3} (e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{3} + 4 \frac{d}{d x} [x^{2}])) - e^{- 3 x^{4} + 4 x^{2}} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5.6

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

$- 5 \frac{x^{3} (e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{3} + 4 (2 x))) - e^{- 3 x^{4} + 4 x^{2}} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5.7

Multiply $2$ by $4$ .

$- 5 \frac{x^{3} (e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{3} + 8 x)) - e^{- 3 x^{4} + 4 x^{2}} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5.8

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

$- 5 \frac{x^{3} (e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{3} + 8 x)) - e^{- 3 x^{4} + 4 x^{2}} (3 x^{2})}{x^{6}}$

Step 5.9

Combine fractions.

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Step 5.9.1

Multiply $3$ by $- 1$ .

$- 5 \frac{x^{3} (e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{3} + 8 x)) - 3 e^{- 3 x^{4} + 4 x^{2}} x^{2}}{x^{6}}$

Step 5.9.2

Combine $- 5$ and $\frac{x^{3} (e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{3} + 8 x)) - 3 e^{- 3 x^{4} + 4 x^{2}} x^{2}}{x^{6}}$ .

$\frac{- 5 (x^{3} (e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{3} + 8 x)) - 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 5.9.3

Move the negative in front of the fraction.

$- \frac{(5) (x^{3} (e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{3} + 8 x)) - 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$- \frac{5 (x^{3} (e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{3} + 8 x))) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2

Simplify the numerator.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Apply the distributive property.

$- \frac{5 (x^{3} (e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{3}) + e^{- 3 x^{4} + 4 x^{2}} (8 x))) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.2

Rewrite using the commutative property of multiplication.

$- \frac{5 (x^{3} (- 12 e^{- 3 x^{4} + 4 x^{2}} x^{3} + e^{- 3 x^{4} + 4 x^{2}} (8 x))) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.3

Rewrite using the commutative property of multiplication.

$- \frac{5 (x^{3} (- 12 e^{- 3 x^{4} + 4 x^{2}} x^{3} + 8 e^{- 3 x^{4} + 4 x^{2}} x)) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.4

Apply the distributive property.

$- \frac{5 (x^{3} (- 12 e^{- 3 x^{4} + 4 x^{2}} x^{3}) + x^{3} (8 e^{- 3 x^{4} + 4 x^{2}} x)) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.5

Rewrite using the commutative property of multiplication.

$- \frac{5 (- 12 x^{3} (e^{- 3 x^{4} + 4 x^{2}} x^{3}) + x^{3} (8 e^{- 3 x^{4} + 4 x^{2}} x)) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.6

Rewrite using the commutative property of multiplication.

$- \frac{5 (- 12 x^{3} (e^{- 3 x^{4} + 4 x^{2}} x^{3}) + 8 x^{3} (e^{- 3 x^{4} + 4 x^{2}} x)) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.7

Simplify each term.

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Step 6.2.1.7.1

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

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Step 6.2.1.7.1.1

Move $x^{3}$ .

$- \frac{5 (- 12 (x^{3} x^{3}) e^{- 3 x^{4} + 4 x^{2}} + 8 x^{3} (e^{- 3 x^{4} + 4 x^{2}} x)) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.7.1.2

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

$- \frac{5 (- 12 x^{3 + 3} e^{- 3 x^{4} + 4 x^{2}} + 8 x^{3} (e^{- 3 x^{4} + 4 x^{2}} x)) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.7.1.3

Add $3$ and $3$ .

$- \frac{5 (- 12 x^{6} e^{- 3 x^{4} + 4 x^{2}} + 8 x^{3} (e^{- 3 x^{4} + 4 x^{2}} x)) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.7.2

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

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Step 6.2.1.7.2.1

Move $x$ .

$- \frac{5 (- 12 x^{6} e^{- 3 x^{4} + 4 x^{2}} + 8 (x \cdot x^{3}) e^{- 3 x^{4} + 4 x^{2}}) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.7.2.2

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

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Step 6.2.1.7.2.2.1

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

$- \frac{5 (- 12 x^{6} e^{- 3 x^{4} + 4 x^{2}} + 8 (x^{1} x^{3}) e^{- 3 x^{4} + 4 x^{2}}) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.7.2.2.2

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

$- \frac{5 (- 12 x^{6} e^{- 3 x^{4} + 4 x^{2}} + 8 x^{1 + 3} e^{- 3 x^{4} + 4 x^{2}}) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.7.2.3

Add $1$ and $3$ .

$- \frac{5 (- 12 x^{6} e^{- 3 x^{4} + 4 x^{2}} + 8 x^{4} e^{- 3 x^{4} + 4 x^{2}}) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.8

Apply the distributive property.

$- \frac{5 (- 12 x^{6} e^{- 3 x^{4} + 4 x^{2}}) + 5 (8 x^{4} e^{- 3 x^{4} + 4 x^{2}}) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.9

Multiply $- 12$ by $5$ .

$- \frac{- 60 (x^{6} e^{- 3 x^{4} + 4 x^{2}}) + 5 (8 x^{4} e^{- 3 x^{4} + 4 x^{2}}) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.10

Multiply $8$ by $5$ .

$- \frac{- 60 (x^{6} e^{- 3 x^{4} + 4 x^{2}}) + 40 (x^{4} e^{- 3 x^{4} + 4 x^{2}}) + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.11

Remove parentheses.

$- \frac{- 60 x^{6} e^{- 3 x^{4} + 4 x^{2}} + 40 x^{4} e^{- 3 x^{4} + 4 x^{2}} + 5 (- 3 e^{- 3 x^{4} + 4 x^{2}} x^{2})}{x^{6}}$

Step 6.2.1.12

Multiply $- 3$ by $5$ .

$- \frac{- 60 x^{6} e^{- 3 x^{4} + 4 x^{2}} + 40 x^{4} e^{- 3 x^{4} + 4 x^{2}} - 15 e^{- 3 x^{4} + 4 x^{2}} x^{2}}{x^{6}}$

Step 6.2.2

Reorder factors in $- 60 x^{6} e^{- 3 x^{4} + 4 x^{2}} + 40 x^{4} e^{- 3 x^{4} + 4 x^{2}} - 15 e^{- 3 x^{4} + 4 x^{2}} x^{2}$ .

$- \frac{- 60 x^{6} e^{- 3 x^{4} + 4 x^{2}} + 40 x^{4} e^{- 3 x^{4} + 4 x^{2}} - 15 x^{2} e^{- 3 x^{4} + 4 x^{2}}}{x^{6}}$

Step 6.3

Factor $5 x^{2} e^{- 3 x^{4} + 4 x^{2}}$ out of $- 60 x^{6} e^{- 3 x^{4} + 4 x^{2}} + 40 x^{4} e^{- 3 x^{4} + 4 x^{2}} - 15 x^{2} e^{- 3 x^{4} + 4 x^{2}}$ .

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Step 6.3.1

Factor $5 x^{2} e^{- 3 x^{4} + 4 x^{2}}$ out of $- 60 x^{6} e^{- 3 x^{4} + 4 x^{2}}$ .

$- \frac{5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4}) + 40 x^{4} e^{- 3 x^{4} + 4 x^{2}} - 15 x^{2} e^{- 3 x^{4} + 4 x^{2}}}{x^{6}}$

Step 6.3.2

Factor $5 x^{2} e^{- 3 x^{4} + 4 x^{2}}$ out of $40 x^{4} e^{- 3 x^{4} + 4 x^{2}}$ .

$- \frac{5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4}) + 5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (8 x^{2}) - 15 x^{2} e^{- 3 x^{4} + 4 x^{2}}}{x^{6}}$

Step 6.3.3

Factor $5 x^{2} e^{- 3 x^{4} + 4 x^{2}}$ out of $- 15 x^{2} e^{- 3 x^{4} + 4 x^{2}}$ .

$- \frac{5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4}) + 5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (8 x^{2}) + 5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (- 3)}{x^{6}}$

Step 6.3.4

Factor $5 x^{2} e^{- 3 x^{4} + 4 x^{2}}$ out of $5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4}) + 5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (8 x^{2})$ .

$- \frac{5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4} + 8 x^{2}) + 5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (- 3)}{x^{6}}$

Step 6.3.5

Factor $5 x^{2} e^{- 3 x^{4} + 4 x^{2}}$ out of $5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4} + 8 x^{2}) + 5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (- 3)$ .

$- \frac{5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4} + 8 x^{2} - 3)}{x^{6}}$

Step 6.4

Cancel the common factor of $x^{2}$ and $x^{6}$ .

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Step 6.4.1

Factor $x^{2}$ out of $5 x^{2} e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4} + 8 x^{2} - 3)$ .

$- \frac{x^{2} (5 e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4} + 8 x^{2} - 3))}{x^{6}}$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $x^{2}$ out of $x^{6}$ .

$- \frac{x^{2} (5 e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4} + 8 x^{2} - 3))}{x^{2} x^{4}}$

Step 6.4.2.2

Cancel the common factor.

$- \frac{x^{2} (5 e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4} + 8 x^{2} - 3))}{x^{2} x^{4}}$

Step 6.4.2.3

Rewrite the expression.

$- \frac{5 e^{- 3 x^{4} + 4 x^{2}} (- 12 x^{4} + 8 x^{2} - 3)}{x^{4}}$

Step 6.5

Factor $- 1$ out of $- 12 x^{4}$ .

$- \frac{5 e^{- 3 x^{4} + 4 x^{2}} (- (12 x^{4}) + 8 x^{2} - 3)}{x^{4}}$

Step 6.6

Factor $- 1$ out of $8 x^{2}$ .

$- \frac{5 e^{- 3 x^{4} + 4 x^{2}} (- (12 x^{4}) - (- 8 x^{2}) - 3)}{x^{4}}$

Step 6.7

Factor $- 1$ out of $- (12 x^{4}) - (- 8 x^{2})$ .

$- \frac{5 e^{- 3 x^{4} + 4 x^{2}} (- (12 x^{4} - 8 x^{2}) - 3)}{x^{4}}$

Step 6.8

Rewrite $- 3$ as $- 1 (3)$ .

$- \frac{5 e^{- 3 x^{4} + 4 x^{2}} (- (12 x^{4} - 8 x^{2}) - 1 (3))}{x^{4}}$

Step 6.9

Factor $- 1$ out of $- (12 x^{4} - 8 x^{2}) - 1 (3)$ .

$- \frac{5 e^{- 3 x^{4} + 4 x^{2}} (- (12 x^{4} - 8 x^{2} + 3))}{x^{4}}$

Step 6.10

Rewrite $- (12 x^{4} - 8 x^{2} + 3)$ as $- 1 (12 x^{4} - 8 x^{2} + 3)$ .

$- \frac{5 e^{- 3 x^{4} + 4 x^{2}} (- 1 (12 x^{4} - 8 x^{2} + 3))}{x^{4}}$

Step 6.11

Move the negative in front of the fraction.

$- - \frac{(5 e^{- 3 x^{4} + 4 x^{2}}) (12 x^{4} - 8 x^{2} + 3)}{x^{4}}$

Step 6.12

Multiply $- 1$ by $- 1$ .

$1 \frac{(5 e^{- 3 x^{4} + 4 x^{2}}) (12 x^{4} - 8 x^{2} + 3)}{x^{4}}$

Step 6.13

Multiply $\frac{(5 e^{- 3 x^{4} + 4 x^{2}}) (12 x^{4} - 8 x^{2} + 3)}{x^{4}}$ by $1$ .

$\frac{(5 e^{- 3 x^{4} + 4 x^{2}}) (12 x^{4} - 8 x^{2} + 3)}{x^{4}}$

Step 6.14

Reorder factors in $\frac{(5 e^{- 3 x^{4} + 4 x^{2}}) (12 x^{4} - 8 x^{2} + 3)}{x^{4}}$ .

$\frac{5 e^{- 3 x^{4} + 4 x^{2}} (12 x^{4} - 8 x^{2} + 3)}{x^{4}}$