91Ó°ÊÓ

A fundamental concept in calculus is the difference quotient, which is crucial for finding derivatives. It essentially measures how a function changes as its input changes by a tiny amount. This concept forms the backbone of derivative calculation. For any given function \( f(x) \), the difference quotient is given by:

• \( \frac{f(x+h) - f(x)}{h} \)

Here, \( h \) is a small, non-zero number representing a tiny increment from \( x \). The numerator \( f(x+h) - f(x) \) represents the change in function value, and dividing by \( h \) scales this change to reflect a rate of change. To find a derivative, we approach the situation where \( h \) becomes infinitesimally small, which leads us to the next concept: limits. Once simplified and evaluated as \( h \) approaches zero, this expression gives us the instantaneous rate of change, or the derivative, of the function at that point.

The binomial theorem is a powerful tool that helps expand expressions raised to a power. In our exercise, we had \( (x+h)^4 \), which requires expansion to simplify the difference quotient.
The binomial theorem states that for any non-negative integer \( n \):

• \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\)

This formula allows us to break down the power of a sum into a series of terms. Each term involves binomial coefficients \( \binom{n}{k} \), which can be calculated using "n choose k," referring to the number of ways to choose k elements from a total of n without regard to order.
In our function, expanding \( (x+h)^4 \) gives us a polynomial:

• \( x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \)

This breakdown allows us to carefully manipulate each term, preparing it for substitution back into the difference quotient.

The limit process is the final and critical step in finding the derivative using the difference quotient. When we calculate \(\lim_{h \to 0} \left(\frac{f(x+h) - f(x)}{h}\right),\)we evaluate how the expression behaves as \( h \) gets closer and closer to zero. This procedure is what we call finding the limit.
For our function \( f(x) = x^4 \), after expanding and simplifying the difference quotient, we had:

• \( 4x^3 + 6x^2h + 4xh^2 + h^3 \)

All terms that include \( h \) become negligible as \( h \) approaches zero:

• \( \lim_{h \to 0} (6x^2h + 4xh^2 + h^3) = 0 \)

Thus, the expression simplifies to just \( 4x^3 \). The limit process removes the influence of the vanishing small parts, yielding the derivative, \( f'(x) = 4x^3 \). This powerful concept is essential because it lets us derive the exact rate of change of a function at any point \( x \).