91Ó°ÊÓ

In calculus, a derivative measures how a function changes as its input changes. It is essentially the slope of the function at a particular point. Let's take the example of the function given:
For the function \( y = x^2 - 1 \), the derivative, denoted \( f'(x) \), shows us the instantaneous rate of change of the function with respect to \( x \). To find the derivative, you apply the definition by calculating the limit:
- Substitute \( f(x+h) \) into the formula.- Simplify the expression.- Factor and cancel terms if possible.- Evaluate the limit as \( h \) approaches 0.
By working through these steps, for \( y = x^2 - 1 \), the result is \( f'(x) = 2x \). This means for any point on the parabola \( y = x^2 - 1 \), the slope of the tangent line at that point is \( 2x \).

Limits are foundational in calculus, providing the basis for defining derivatives, integrals, and continuity. A limit captures the behavior of a function as its input approaches a particular value.
When taking the derivative, we compute:\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]This limit represents the idea of considering the slope as \( h \), a small increase in \( x \), approaches zero.
It is important to note that as \( h \) approaches zero, we never actually allow \( h \) to be zero since the expression \( \frac{f(x+h) - f(x)}{h} \) would be undefined. Instead, we explore values of \( h \) that are exceedingly small to perceive the trend or behavior of the function.

The binomial formula is a mathematical tool that is often used to expand expressions raised to a power. In calculus, it assists in the computation of derivatives, particularly when dealing with polynomial functions.
For example, consider the expression \( (x+h)^2 \), which appears in the process of finding derivatives:- By expanding, the expression becomes \( x^2 + 2xh + h^2 \).- This expansion is derived using the binomial theorem, which states:\[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]In our case, \( a = x \), \( b = h \), and \( n = 2 \). The expanded form helps us simplify our expressions when calculating limits and derivatives.

Function analysis involves examining the behavior, properties, and structure of mathematical functions. It's a broad field in calculus that includes understanding how a function behaves at various points and intervals.
For the function \( y = x^2 - 1 \), we already know:- It is a quadratic function represented by a parabola.- The vertex lies at (0, -1), serving as the minimum point.
Analyzing its derivative, \( f'(x) = 2x \), provides insights into:- **Slope**: The slope is positive when \( x > 0 \) and negative when \( x < 0 \), indicating the parabola opens upwards.- **Intervals of Increase/Decrease**: The function is increasing for \( x > 0 \) and decreasing for \( x < 0 \).Function analysis aids in deeper understanding, not only predicting function behavior but also optimizing and solving real-world problems involving motion, growth, and more.