Calculus Examples

$h (x) = \frac{3 x^{2} - 2 e^{x}}{5 x}$

Step 1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{3 x^{2} - 2 e^{x}}{5 x}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [\frac{3 x^{2} - 2 e^{x}}{x}]$ .

$\frac{1}{5} \frac{d}{d x} [\frac{3 x^{2} - 2 e^{x}}{x}]$

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

$\frac{1}{5} \cdot \frac{x \frac{d}{d x} [3 x^{2} - 2 e^{x}] - (3 x^{2} - 2 e^{x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$\frac{1}{5} \cdot \frac{x (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 2 e^{x}]) - (3 x^{2} - 2 e^{x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2

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

$\frac{1}{5} \cdot \frac{x (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 e^{x}]) - (3 x^{2} - 2 e^{x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3

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

$\frac{1}{5} \cdot \frac{x (3 (2 x) + \frac{d}{d x} [- 2 e^{x}]) - (3 x^{2} - 2 e^{x}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.4

Multiply $2$ by $3$ .

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

Step 3.5

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

$\frac{1}{5} \cdot \frac{x (6 x - 2 \frac{d}{d x} [e^{x}]) - (3 x^{2} - 2 e^{x}) \frac{d}{d x} [x]}{x^{2}}$

Step 4

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

$\frac{1}{5} \cdot \frac{x (6 x - 2 e^{x}) - (3 x^{2} - 2 e^{x}) \frac{d}{d x} [x]}{x^{2}}$

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

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

Step 5.2

Combine fractions.

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Step 5.2.1

Multiply $- 1$ by $1$ .

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

Step 5.2.2

Multiply $\frac{1}{5}$ by $\frac{x (6 x - 2 e^{x}) - (3 x^{2} - 2 e^{x})}{x^{2}}$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.3.1.2

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

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Step 6.3.1.2.1

Move $x$ .

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

Step 6.3.1.2.2

Multiply $x$ by $x$ .

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

Step 6.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 6.3.1.4

Multiply $3$ by $- 1$ .

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

Step 6.3.1.5

Multiply $- 2$ by $- 1$ .

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

Step 6.3.2

Subtract $3 x^{2}$ from $6 x^{2}$ .

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

Step 6.4

Reorder terms.

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

Step 6.5

Reorder factors in $\frac{3 x^{2} - 2 e^{x} x + 2 e^{x}}{5 x^{2}}$ .

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