91Ó°ÊÓ

The concept of a limit is foundational in calculus and is essential to understanding how derivatives are calculated. The limit definition of a derivative is expressed as: \[ f'(x) = \lim _{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} \] This definition represents the instantaneous rate of change of the function \( f(x) \) at a specific point \( x \). The variable \( h \) represents an infinitesimally small change in \( x \), allowing us to estimate how \( f(x) \) behaves close to \( x \). By taking \( h \) to approach zero, we can more accurately calculate the derivative. In the example, when finding the derivative of \( f(x) = 2x^2 + 5 \), the goal is to see how the function's value changes as \( x \) slightly increases by \( h \), and the limit ensures that we examine the change at the exact point.

The difference quotient is a central part of the limit definition of the derivative. It helps to express the average rate of change of the function \( f(x) \) over the interval \( [x, x+h] \). In our example, the difference quotient is set up as: \[ \frac{f(x+h) - f(x)}{h} \] This expression allows us to compute the slope of the secant line passing through the points \( (x, f(x)) \) and \( (x+h, f(x+h)) \). By plugging \( f(x) = 2x^2 + 5 \) into the difference quotient, you get:

• Substitute \( f(x+h) = 2(x+h)^2 + 5 \): Expand and simplify to obtain \( f(x+h) = 2x^2 + 4xh + 2h^2 + 5 \).
• Use the difference quotient formula to get \( \frac{(2x^2 + 4xh + 2h^2 + 5) - (2x^2 + 5)}{h} \).
• Simplify this to \( \frac{4xh + 2h^2}{h} \).
• Cancel out \( h \) to get \( 4x + 2h \).

This process transforms the average rate of change over the interval into an instantaneous rate of change as you proceed to take the limit.

Polynomial functions are relatively straightforward when it comes to finding derivatives, and they provide a good opportunity to practice applying the limit definition and using difference quotients. In our example, the function \( f(x) = 2x^2 + 5 \) is a polynomial, specifically a quadratic function. Taking its derivative involves: - Observing how the terms of the polynomial are affected by the derivative process. Each term in the linear form of \( ax^n \) becomes \( nax^{n-1} \) when derived.

• The term \( 2x^2 \) becomes \( 4x \) because \( n = 2 \) and you bring down the power as a coefficient (\( 2 \times 2 \)) and reduce the exponent by one.
• The constant term \( 5 \) becomes zero, as constants do not change when their rate of change is calculated.

Thus, applying these rules and confirming via the limit definition gives us the final derivative: \( f'(x) = 4x \). Because polynomial functions follow these simple derivative rules, once you understand the process, you can easily compute derivatives of more complex polynomials, allowing you to efficiently determine how a polynomial behaves at any given point.