Precalculus Examples

$y = a {(2)}^{\frac{t}{3}}$

Step 1

Write $y = a {(2)}^{\frac{t}{3}}$ as a function.

$f (a) = a {(2)}^{\frac{t}{3}}$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

$f (a + h) = (a + h) {(2)}^{\frac{t}{3}}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

$f (a + h) = a \cdot 2^{\frac{t}{3}} + h \cdot 2^{\frac{t}{3}}$

Step 3.1.2.2

The final answer is $a \cdot 2^{\frac{t}{3}} + h \cdot 2^{\frac{t}{3}}$ .

$a \cdot 2^{\frac{t}{3}} + h \cdot 2^{\frac{t}{3}}$

Step 3.2

Reorder.

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Step 3.2.1

Reorder $a$ and $2^{\frac{t}{3}}$ .

$2^{\frac{t}{3}} \cdot a + h \cdot 2^{\frac{t}{3}}$

Step 3.2.2

Reorder $h$ and $2^{\frac{t}{3}}$ .

$2^{\frac{t}{3}} \cdot a + 2^{\frac{t}{3}} \cdot h$

Step 3.3

Find the components of the definition.

$f (a + h) = 2^{\frac{t}{3}} \cdot a + 2^{\frac{t}{3}} \cdot h$

$f (a) = a \cdot 2^{\frac{t}{3}}$

$f (a + h) = 2^{\frac{t}{3}} \cdot a + 2^{\frac{t}{3}} \cdot h$

$f (a) = a \cdot 2^{\frac{t}{3}}$

Step 4

Plug in the components.

$\frac{f (a + h) - f a}{h} = \frac{2^{\frac{t}{3}} \cdot a + 2^{\frac{t}{3}} \cdot h - (a \cdot 2^{\frac{t}{3}})}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Subtract $a \cdot 2^{\frac{t}{3}}$ from $2^{\frac{t}{3}} a$ .

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Step 5.1.1.1

Move $a$ .

$\frac{2^{\frac{t}{3}} h + 2^{\frac{t}{3}} a - 1 \cdot 2^{\frac{t}{3}} a}{h}$

Step 5.1.1.2

Subtract $2^{\frac{t}{3}} a$ from $2^{\frac{t}{3}} a$ .

$\frac{2^{\frac{t}{3}} h + 0}{h}$

Step 5.1.2

Add $2^{\frac{t}{3}} h$ and $0$ .

$\frac{2^{\frac{t}{3}} h}{h}$

Step 5.2

Cancel the common factor of $h$ .

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Step 5.2.1

Cancel the common factor.

$\frac{2^{\frac{t}{3}} h}{h}$

Step 5.2.2

Divide $2^{\frac{t}{3}}$ by $1$ .

$2^{\frac{t}{3}}$

Step 6