Consider a function $y=f\left(x\right)$. Which of the following best defines the slope of this function at a point $x=a$?

• A. The value of the function $f\left(a\right)$
• B. The value of the derivative $f^{\prime }\left(a\right)$
• C. The value of $x$ at that point
• D. The area under the curve from $0$ to $a$

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the concept of the slope of a function at a point. The slope at a specific point on a curve is defined as the rate of change of the function at that point. This is mathematically represented by the derivative of the function at that point.
2. Step 2: Recall the definition of the derivative. The derivative of a function f(x) at a point x = a, denoted as f'(a), is the limit of the difference quotient as the interval approaches zero: $f\text{'}(a)\; =\; lim\_(h\; \to \; 0)\; [(f(a\; +\; h)\; -\; f(a))/h]$.
3. Step 3: Compare the options provided in the problem. The value of the function f(a) represents the height of the curve at x = a, not the slope. The value of x at that point is simply the x-coordinate and does not describe the slope. The area under the curve from 0 to a is related to integration, not differentiation.
4. Step 4: Recognize that the correct answer is the value of the derivative f'(a), as it directly defines the slope of the function at x = a.
5. Step 5: Conclude that the slope of the function y = f(x) at a point x = a is best defined by the value of the derivative f'(a).