Precalculus Examples

$y = 2 x e^{3 x}$

Step 1

Write $y = 2 x e^{3 x}$ as a function.

$f (x) = 2 x e^{3 x}$

Step 2

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 3

Find the components of the definition.

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Step 3.1

Evaluate the function at $x = x + h$ .

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Step 3.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 2 (x + h) \cdot e^{3 (x + h)}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

$f (x + h) = (2 x + 2 h) \cdot e^{3 (x + h)}$

Step 3.1.2.2

Apply the distributive property.

$f (x + h) = (2 x + 2 h) \cdot e^{3 x + 3 h}$

Step 3.1.2.3

Apply the distributive property.

$f (x + h) = 2 x e^{3 x + 3 h} + 2 h e^{3 x + 3 h}$

Step 3.1.2.4

The final answer is $2 x e^{3 x + 3 h} + 2 h e^{3 x + 3 h}$ .

$2 x e^{3 x + 3 h} + 2 h e^{3 x + 3 h}$

Step 3.2

Find the components of the definition.

$f (x + h) = 2 x e^{3 x + 3 h} + 2 h e^{3 x + 3 h}$

$f (x) = 2 x e^{3 x}$

$f (x + h) = 2 x e^{3 x + 3 h} + 2 h e^{3 x + 3 h}$

$f (x) = 2 x e^{3 x}$

Step 4

Plug in the components.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $2$ .

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

Step 5.2

Factor $2$ out of $2 x e^{3 x + 3 h} + 2 h e^{3 x + 3 h} - 2 x e^{3 x}$ .

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Step 5.2.1

Factor $2$ out of $2 x e^{3 x + 3 h}$ .

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

Step 5.2.2

Factor $2$ out of $2 h e^{3 x + 3 h}$ .

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

Step 5.2.3

Factor $2$ out of $- 2 x e^{3 x}$ .

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

Step 5.2.4

Factor $2$ out of $2 (x e^{3 x + 3 h}) + 2 (h e^{3 x + 3 h})$ .

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

Step 5.2.5

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

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

Step 6