Calculus Examples

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

Step 1

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 $f (x) = 3 x^{2} + 2 e^{x}$ and $g (x) = 5 e^{x} - 4 x$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $3$ .

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

Step 2.5

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

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

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $5 e^{x} - 4 x$ with respect to $x$ is $\frac{d}{d x} [5 e^{x}] + \frac{d}{d x} [- 4 x]$ .

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

Step 4.2

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

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

Step 5

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

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

Step 6

Differentiate.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Multiply $- 4$ by $1$ .

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Simplify the numerator.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Expand $(5 e^{x} - 4 x) (6 x + 2 e^{x})$ using the FOIL Method.

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Step 7.2.1.1.1

Apply the distributive property.

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

Step 7.2.1.1.2

Apply the distributive property.

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

Step 7.2.1.1.3

Apply the distributive property.

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

Step 7.2.1.2

Simplify and combine like terms.

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Step 7.2.1.2.1

Simplify each term.

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Step 7.2.1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 7.2.1.2.1.2

Multiply $5$ by $6$ .

$\frac{30 e^{x} x + 5 e^{x} (2 e^{x}) - 4 x (6 x) - 4 x (2 e^{x}) + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.1.3

Rewrite using the commutative property of multiplication.

$\frac{30 e^{x} x + 5 \cdot 2 e^{x} e^{x} - 4 x (6 x) - 4 x (2 e^{x}) + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.1.4

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

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Step 7.2.1.2.1.4.1

Move $e^{x}$ .

$\frac{30 e^{x} x + 5 \cdot 2 (e^{x} e^{x}) - 4 x (6 x) - 4 x (2 e^{x}) + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.1.4.2

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

$\frac{30 e^{x} x + 5 \cdot 2 e^{x + x} - 4 x (6 x) - 4 x (2 e^{x}) + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.1.4.3

Add $x$ and $x$ .

$\frac{30 e^{x} x + 5 \cdot 2 e^{2 x} - 4 x (6 x) - 4 x (2 e^{x}) + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.1.5

Multiply $5$ by $2$ .

$\frac{30 e^{x} x + 10 e^{2 x} - 4 x (6 x) - 4 x (2 e^{x}) + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.1.6

Rewrite using the commutative property of multiplication.

$\frac{30 e^{x} x + 10 e^{2 x} - 4 \cdot 6 x \cdot x - 4 x (2 e^{x}) + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.1.7

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

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Step 7.2.1.2.1.7.1

Move $x$ .

$\frac{30 e^{x} x + 10 e^{2 x} - 4 \cdot 6 (x \cdot x) - 4 x (2 e^{x}) + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.1.7.2

Multiply $x$ by $x$ .

$\frac{30 e^{x} x + 10 e^{2 x} - 4 \cdot 6 x^{2} - 4 x (2 e^{x}) + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.1.8

Multiply $- 4$ by $6$ .

$\frac{30 e^{x} x + 10 e^{2 x} - 24 x^{2} - 4 x (2 e^{x}) + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.1.9

Rewrite using the commutative property of multiplication.

$\frac{30 e^{x} x + 10 e^{2 x} - 24 x^{2} - 4 \cdot 2 x e^{x} + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.1.10

Multiply $- 4$ by $2$ .

$\frac{30 e^{x} x + 10 e^{2 x} - 24 x^{2} - 8 x e^{x} + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.2

Subtract $8 x e^{x}$ from $30 e^{x} x$ .

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Step 7.2.1.2.2.1

Move $e^{x}$ .

$\frac{30 x e^{x} - 8 x e^{x} + 10 e^{2 x} - 24 x^{2} + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.2.2.2

Subtract $8 x e^{x}$ from $30 x e^{x}$ .

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} + (- (3 x^{2}) - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.3

Simplify each term.

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Step 7.2.1.3.1

Multiply $3$ by $- 1$ .

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} + (- 3 x^{2} - (2 e^{x})) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.3.2

Multiply $2$ by $- 1$ .

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} + (- 3 x^{2} - 2 e^{x}) (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.4

Expand $(- 3 x^{2} - 2 e^{x}) (5 e^{x} - 4)$ using the FOIL Method.

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Step 7.2.1.4.1

Apply the distributive property.

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 3 x^{2} (5 e^{x} - 4) - 2 e^{x} (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.4.2

Apply the distributive property.

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 3 x^{2} (5 e^{x}) - 3 x^{2} \cdot - 4 - 2 e^{x} (5 e^{x} - 4)}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.4.3

Apply the distributive property.

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 3 x^{2} (5 e^{x}) - 3 x^{2} \cdot - 4 - 2 e^{x} (5 e^{x}) - 2 e^{x} \cdot - 4}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.5

Simplify each term.

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Step 7.2.1.5.1

Rewrite using the commutative property of multiplication.

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 3 \cdot 5 x^{2} e^{x} - 3 x^{2} \cdot - 4 - 2 e^{x} (5 e^{x}) - 2 e^{x} \cdot - 4}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.5.2

Multiply $- 3$ by $5$ .

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 15 x^{2} e^{x} - 3 x^{2} \cdot - 4 - 2 e^{x} (5 e^{x}) - 2 e^{x} \cdot - 4}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.5.3

Multiply $- 4$ by $- 3$ .

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 15 x^{2} e^{x} + 12 x^{2} - 2 e^{x} (5 e^{x}) - 2 e^{x} \cdot - 4}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.5.4

Rewrite using the commutative property of multiplication.

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 15 x^{2} e^{x} + 12 x^{2} - 2 \cdot 5 e^{x} e^{x} - 2 e^{x} \cdot - 4}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.5.5

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

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Step 7.2.1.5.5.1

Move $e^{x}$ .

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 15 x^{2} e^{x} + 12 x^{2} - 2 \cdot 5 (e^{x} e^{x}) - 2 e^{x} \cdot - 4}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.5.5.2

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

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 15 x^{2} e^{x} + 12 x^{2} - 2 \cdot 5 e^{x + x} - 2 e^{x} \cdot - 4}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.5.5.3

Add $x$ and $x$ .

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 15 x^{2} e^{x} + 12 x^{2} - 2 \cdot 5 e^{2 x} - 2 e^{x} \cdot - 4}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.5.6

Multiply $- 2$ by $5$ .

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 15 x^{2} e^{x} + 12 x^{2} - 10 e^{2 x} - 2 e^{x} \cdot - 4}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.1.5.7

Multiply $- 4$ by $- 2$ .

$\frac{22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 15 x^{2} e^{x} + 12 x^{2} - 10 e^{2 x} + 8 e^{x}}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.2

Combine the opposite terms in $22 x e^{x} + 10 e^{2 x} - 24 x^{2} - 15 x^{2} e^{x} + 12 x^{2} - 10 e^{2 x} + 8 e^{x}$ .

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Step 7.2.2.1

Subtract $10 e^{2 x}$ from $10 e^{2 x}$ .

$\frac{22 x e^{x} - 24 x^{2} - 15 x^{2} e^{x} + 12 x^{2} + 0 + 8 e^{x}}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.2.2

Add $22 x e^{x} - 24 x^{2} - 15 x^{2} e^{x} + 12 x^{2}$ and $0$ .

$\frac{22 x e^{x} - 24 x^{2} - 15 x^{2} e^{x} + 12 x^{2} + 8 e^{x}}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.2.3

Add $- 24 x^{2}$ and $12 x^{2}$ .

$\frac{22 x e^{x} - 12 x^{2} - 15 x^{2} e^{x} + 8 e^{x}}{{(5 e^{x} - 4 x)}^{2}}$

Step 7.3

Reorder terms.

$\frac{- 12 x^{2} + 22 e^{x} x - 15 e^{x} x^{2} + 8 e^{x}}{{(- 4 x + 5 e^{x})}^{2}}$

Step 7.4

Factor $- 1$ out of $- 12 x^{2}$ .

$\frac{- (12 x^{2}) + 22 e^{x} x - 15 e^{x} x^{2} + 8 e^{x}}{{(- 4 x + 5 e^{x})}^{2}}$

Step 7.5

Factor $- 1$ out of $22 e^{x} x$ .

$\frac{- (12 x^{2}) - (- 22 e^{x} x) - 15 e^{x} x^{2} + 8 e^{x}}{{(- 4 x + 5 e^{x})}^{2}}$

Step 7.6

Factor $- 1$ out of $- (12 x^{2}) - (- 22 e^{x} x)$ .

$\frac{- (12 x^{2} - 22 e^{x} x) - 15 e^{x} x^{2} + 8 e^{x}}{{(- 4 x + 5 e^{x})}^{2}}$

Step 7.7

Factor $- 1$ out of $- 15 e^{x} x^{2}$ .

$\frac{- (12 x^{2} - 22 e^{x} x) - (15 e^{x} x^{2}) + 8 e^{x}}{{(- 4 x + 5 e^{x})}^{2}}$

Step 7.8

Factor $- 1$ out of $- (12 x^{2} - 22 e^{x} x) - (15 e^{x} x^{2})$ .

$\frac{- (12 x^{2} - 22 e^{x} x + 15 e^{x} x^{2}) + 8 e^{x}}{{(- 4 x + 5 e^{x})}^{2}}$

Step 7.9

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

$\frac{- (12 x^{2} - 22 e^{x} x + 15 e^{x} x^{2}) - (- 8 e^{x})}{{(- 4 x + 5 e^{x})}^{2}}$

Step 7.10

Factor $- 1$ out of $- (12 x^{2} - 22 e^{x} x + 15 e^{x} x^{2}) - (- 8 e^{x})$ .

$\frac{- (12 x^{2} - 22 e^{x} x + 15 e^{x} x^{2} - 8 e^{x})}{{(- 4 x + 5 e^{x})}^{2}}$

Step 7.11

Rewrite $- (12 x^{2} - 22 e^{x} x + 15 e^{x} x^{2} - 8 e^{x})$ as $- 1 (12 x^{2} - 22 e^{x} x + 15 e^{x} x^{2} - 8 e^{x})$ .

$\frac{- 1 (12 x^{2} - 22 e^{x} x + 15 e^{x} x^{2} - 8 e^{x})}{{(- 4 x + 5 e^{x})}^{2}}$

Step 7.12

Move the negative in front of the fraction.

$- \frac{12 x^{2} - 22 e^{x} x + 15 e^{x} x^{2} - 8 e^{x}}{{(- 4 x + 5 e^{x})}^{2}}$

Step 7.13

Reorder factors in $- \frac{12 x^{2} - 22 e^{x} x + 15 e^{x} x^{2} - 8 e^{x}}{{(- 4 x + 5 e^{x})}^{2}}$ .

$- \frac{12 x^{2} - 22 x e^{x} + 15 x^{2} e^{x} - 8 e^{x}}{{(- 4 x + 5 e^{x})}^{2}}$