The given expression is:

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

This is a fundamental expression in calculus, particularly when discussing derivatives. It represents the difference quotient, which is the foundation for determining the rate of change of a function.

Meaning:

1. $f(x)$: Value of the function at $x$.
2. $f(x+h)$: Value of the function at a point slightly "ahead" of $x$ by $h$, where $h$ is a small increment.
3. $h$: The change in $x$ (step size or increment).

Limit Definition of the Derivative:

As $h \to 0$, the difference quotient becomes the derivative of $f(x)$. Mathematically:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• $f'(x)$ gives the instantaneous rate of change or the slope of the tangent line to the curve $f(x)$ at a given point $x$.

Would you like me to go deeper into derivatives, limits, or any specific example?

Related Questions:

1. What is the physical significance of the derivative?
2. How do you compute the derivative of a polynomial using this formula?
3. What happens when $h$ is not small enough in numerical calculations?
4. Can this formula be extended to higher-order derivatives?
5. How do you interpret this formula in terms of a graph?

Tip:

The derivative formula helps analyze rates of change, such as velocity, acceleration, and slopes of curves. It is essential in physics, engineering, and economics.