Given the function $f(x) = 5^x$, we want to find the expression for the difference quotient:

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

Step-by-Step Calculation:

1. Substitute the Function:

   Substitute $f(x) = 5^x$ into the difference quotient:

   $\frac{5^{x+h} - 5^x}{h}$

2. Apply Properties of Exponents:

   Recall that $5^{x+h} = 5^x \cdot 5^h$. Using this, we rewrite the expression:

   $\frac{5^x \cdot 5^h - 5^x}{h}$

3. Factor Out Common Terms:

   Factor out $5^x$ from the numerator:

   $= \frac{5^x (5^h - 1)}{h}$

Simplified Difference Quotient:

The difference quotient for $f(x) = 5^x$ is:

$\frac{f(x+h) - f(x)}{h} = \frac{5^x (5^h - 1)}{h}$

Interpretation:

• This expression represents the average rate of change of $5^x$ over an interval of $h$.
• If you take the limit as $h \to 0$, this expression will give the derivative of $f(x) = 5^x$.

Would you like to see the derivative calculation using this quotient?

Related Questions:

1. How do you find the derivative of $5^x$ using limits?
2. Why is the difference quotient important in calculus?
3. How does the expression change if $f(x) = a^x$ for a different base $a$?
4. What is the derivative of exponential functions in general?
5. What happens to the difference quotient as $h \to 0$?

Tip: Knowing the difference quotient is fundamental in understanding how a function changes, leading to the concept of the derivative in calculus.