Calculus Examples

$\ln (\frac{e^{x^{3}}}{x^{4} - 7 x + 1})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{e^{x^{3}}}{x^{4} - 7 x + 1}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{e^{x^{3}}}{x^{4} - 7 x + 1}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\frac{e^{x^{3}}}{x^{4} - 7 x + 1}]$

Step 1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [\frac{e^{x^{3}}}{x^{4} - 7 x + 1}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\frac{e^{x^{3}}}{x^{4} - 7 x + 1}$ .

$\frac{1}{\frac{e^{x^{3}}}{x^{4} - 7 x + 1}} \frac{d}{d x} [\frac{e^{x^{3}}}{x^{4} - 7 x + 1}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{e^{x^{3}}}{x^{4} - 7 x + 1}$ .

$1 \frac{x^{4} - 7 x + 1}{e^{x^{3}}} \frac{d}{d x} [\frac{e^{x^{3}}}{x^{4} - 7 x + 1}]$

Step 3

Multiply $\frac{x^{4} - 7 x + 1}{e^{x^{3}}}$ by $1$ .

$\frac{x^{4} - 7 x + 1}{e^{x^{3}}} \frac{d}{d x} [\frac{e^{x^{3}}}{x^{4} - 7 x + 1}]$

Step 4

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 $f (x) = e^{x^{3}}$ and $g (x) = x^{4} - 7 x + 1$ .

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

Step 5

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 $g (x) = x^{3}$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $x^{3}$ .

$\frac{x^{4} - 7 x + 1}{e^{x^{3}}} \cdot \frac{(x^{4} - 7 x + 1) (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [x^{3}]) - e^{x^{3}} \frac{d}{d x} [x^{4} - 7 x + 1]}{{(x^{4} - 7 x + 1)}^{2}}$

Step 5.2

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

$\frac{x^{4} - 7 x + 1}{e^{x^{3}}} \cdot \frac{(x^{4} - 7 x + 1) (e^{u_{2}} \frac{d}{d x} [x^{3}]) - e^{x^{3}} \frac{d}{d x} [x^{4} - 7 x + 1]}{{(x^{4} - 7 x + 1)}^{2}}$

Step 5.3

Replace all occurrences of $u_{2}$ with $x^{3}$ .

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

Step 6

Differentiate.

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Step 6.1

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

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

Step 6.2

By the Sum Rule, the derivative of $x^{4} - 7 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [1]$ .

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

Multiply $- 7$ by $1$ .

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

Step 6.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{x^{4} - 7 x + 1}{e^{x^{3}}} \cdot \frac{(x^{4} - 7 x + 1) e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3} - 7 + 0)}{{(x^{4} - 7 x + 1)}^{2}}$

Step 6.8

Combine fractions.

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Step 6.8.1

Add $4 x^{3} - 7$ and $0$ .

$\frac{x^{4} - 7 x + 1}{e^{x^{3}}} \cdot \frac{(x^{4} - 7 x + 1) e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3} - 7)}{{(x^{4} - 7 x + 1)}^{2}}$

Step 6.8.2

Multiply $\frac{x^{4} - 7 x + 1}{e^{x^{3}}}$ by $\frac{(x^{4} - 7 x + 1) e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3} - 7)}{{(x^{4} - 7 x + 1)}^{2}}$ .

$\frac{(x^{4} - 7 x + 1) ((x^{4} - 7 x + 1) e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3} - 7))}{e^{x^{3}} {(x^{4} - 7 x + 1)}^{2}}$

Step 7

Cancel the common factors.

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Step 7.1

Factor $x^{4} - 7 x + 1$ out of $e^{x^{3}} {(x^{4} - 7 x + 1)}^{2}$ .

$\frac{(x^{4} - 7 x + 1) ((x^{4} - 7 x + 1) e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3} - 7))}{(x^{4} - 7 x + 1) (e^{x^{3}} (x^{4} - 7 x + 1))}$

Step 7.2

Cancel the common factor.

$\frac{(x^{4} - 7 x + 1) ((x^{4} - 7 x + 1) e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3} - 7))}{(x^{4} - 7 x + 1) (e^{x^{3}} (x^{4} - 7 x + 1))}$

Step 7.3

Rewrite the expression.

$\frac{(x^{4} - 7 x + 1) e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3} - 7)}{e^{x^{3}} (x^{4} - 7 x + 1)}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{(x^{4} e^{x^{3}} - 7 x e^{x^{3}} + 1 e^{x^{3}}) (3 x^{2}) - e^{x^{3}} (4 x^{3} - 7)}{e^{x^{3}} (x^{4} - 7 x + 1)}$

Step 8.2

Apply the distributive property.

$\frac{x^{4} e^{x^{3}} (3 x^{2}) - 7 x e^{x^{3}} (3 x^{2}) + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3} - 7)}{e^{x^{3}} (x^{4} - 7 x + 1)}$

Step 8.3

Apply the distributive property.

$\frac{x^{4} e^{x^{3}} (3 x^{2}) - 7 x e^{x^{3}} (3 x^{2}) + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} (x^{4} - 7 x + 1)}$

Step 8.4

Apply the distributive property.

$\frac{x^{4} e^{x^{3}} (3 x^{2}) - 7 x e^{x^{3}} (3 x^{2}) + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Simplify each term.

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Step 8.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{3 x^{4} e^{x^{3}} x^{2} - 7 x e^{x^{3}} (3 x^{2}) + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.2

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 8.5.1.2.1

Move $x^{2}$ .

$\frac{3 (x^{2} x^{4}) e^{x^{3}} - 7 x e^{x^{3}} (3 x^{2}) + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.2.2

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

$\frac{3 x^{2 + 4} e^{x^{3}} - 7 x e^{x^{3}} (3 x^{2}) + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.2.3

Add $2$ and $4$ .

$\frac{3 x^{6} e^{x^{3}} - 7 x e^{x^{3}} (3 x^{2}) + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.3

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 8.5.1.3.1

Move $x^{2}$ .

$\frac{3 x^{6} e^{x^{3}} - 7 (x^{2} x) e^{x^{3}} \cdot 3 + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.3.2

Multiply $x^{2}$ by $x$ .

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Step 8.5.1.3.2.1

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

$\frac{3 x^{6} e^{x^{3}} - 7 (x^{2} x^{1}) e^{x^{3}} \cdot 3 + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.3.2.2

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

$\frac{3 x^{6} e^{x^{3}} - 7 x^{2 + 1} e^{x^{3}} \cdot 3 + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.3.3

Add $2$ and $1$ .

$\frac{3 x^{6} e^{x^{3}} - 7 x^{3} e^{x^{3}} \cdot 3 + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.4

Multiply $3$ by $- 7$ .

$\frac{3 x^{6} e^{x^{3}} - 21 x^{3} e^{x^{3}} + 1 e^{x^{3}} (3 x^{2}) - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.5

Rewrite using the commutative property of multiplication.

$\frac{3 x^{6} e^{x^{3}} - 21 x^{3} e^{x^{3}} + 1 \cdot 3 e^{x^{3}} x^{2} - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.6

Multiply $3$ by $1$ .

$\frac{3 x^{6} e^{x^{3}} - 21 x^{3} e^{x^{3}} + 3 e^{x^{3}} x^{2} - e^{x^{3}} (4 x^{3}) - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.7

Rewrite using the commutative property of multiplication.

$\frac{3 x^{6} e^{x^{3}} - 21 x^{3} e^{x^{3}} + 3 e^{x^{3}} x^{2} - 1 \cdot 4 e^{x^{3}} x^{3} - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.8

Multiply $- 1$ by $4$ .

$\frac{3 x^{6} e^{x^{3}} - 21 x^{3} e^{x^{3}} + 3 e^{x^{3}} x^{2} - 4 e^{x^{3}} x^{3} - e^{x^{3}} \cdot - 7}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.1.9

Multiply $- 7$ by $- 1$ .

$\frac{3 x^{6} e^{x^{3}} - 21 x^{3} e^{x^{3}} + 3 e^{x^{3}} x^{2} - 4 e^{x^{3}} x^{3} + 7 e^{x^{3}}}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.2

Subtract $4 e^{x^{3}} x^{3}$ from $- 21 x^{3} e^{x^{3}}$ .

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Step 8.5.2.1

Move $e^{x^{3}}$ .

$\frac{3 x^{6} e^{x^{3}} - 21 x^{3} e^{x^{3}} - 4 x^{3} e^{x^{3}} + 3 e^{x^{3}} x^{2} + 7 e^{x^{3}}}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.2.2

Subtract $4 x^{3} e^{x^{3}}$ from $- 21 x^{3} e^{x^{3}}$ .

$\frac{3 x^{6} e^{x^{3}} - 25 x^{3} e^{x^{3}} + 3 e^{x^{3}} x^{2} + 7 e^{x^{3}}}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.5.3

Reorder factors in $3 x^{6} e^{x^{3}} - 25 x^{3} e^{x^{3}} + 3 e^{x^{3}} x^{2} + 7 e^{x^{3}}$ .

$\frac{3 x^{6} e^{x^{3}} - 25 x^{3} e^{x^{3}} + 3 x^{2} e^{x^{3}} + 7 e^{x^{3}}}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}} \cdot 1}$

Step 8.6

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

$\frac{3 x^{6} e^{x^{3}} - 25 x^{3} e^{x^{3}} + 3 x^{2} e^{x^{3}} + 7 e^{x^{3}}}{e^{x^{3}} x^{4} + e^{x^{3}} (- 7 x) + e^{x^{3}}}$

Step 8.7

Reorder terms.

$\frac{3 e^{x^{3}} x^{6} - 25 e^{x^{3}} x^{3} + 3 e^{x^{3}} x^{2} + 7 e^{x^{3}}}{e^{x^{3}} x^{4} - 7 e^{x^{3}} x + e^{x^{3}}}$