Evaluate The Derivative Of The Function At The Given Point.

Evaluating the Derivative of a Function at a Given Point

Determining the derivative of a function at a specific point is a fundamental concept in calculus with wide-ranging applications in various fields. This process allows us to understand the instantaneous rate of change of a function at that precise point, providing valuable insights into its behavior. This article will delve into the various methods for evaluating derivatives at a given point, exploring both theoretical underpinnings and practical examples. We'll cover the definition of the derivative, different techniques for finding derivatives, and applications of these concepts.

Understanding the Derivative

Before we jump into evaluating derivatives at specific points, let's solidify our understanding of the derivative itself. The derivative of a function, often denoted as f'(x) or df/dx, represents the instantaneous rate of change of the function at any given point x. Geometrically, it represents the slope of the tangent line to the function's graph at that point.

The derivative is formally defined using the limit of a difference quotient:

f'(x) = lim (h→0) [(f(x + h) - f(x)) / h]

This definition captures the essence of the instantaneous rate of change: as the change in x (h) approaches zero, the difference quotient approaches the slope of the tangent line at x.

Key Concepts:

• Difference Quotient: The expression [(f(x + h) - f(x)) / h] represents the average rate of change of the function over a small interval h.
• Limit: The limit operation is crucial; it allows us to approach an infinitely small interval, giving us the instantaneous rate of change.
• Instantaneous Rate of Change: This represents the precise rate of change at a single point, unlike the average rate of change over an interval.

Methods for Evaluating Derivatives

Several methods exist for finding the derivative of a function, each with its own strengths and weaknesses. Let's explore some of the most common techniques:

1. The Limit Definition (First Principles)

This method directly applies the limit definition of the derivative. While conceptually important, it can be cumbersome for complex functions.

Example: Let's find the derivative of f(x) = x² at x = 2 using the limit definition:

1. Substitute into the difference quotient: [( (x + h)² - x²) / h]
2. Expand and simplify: [(x² + 2xh + h² - x²) / h] = [2xh + h²] / h = 2x + h
3. Take the limit as h approaches 0: lim (h→0) (2x + h) = 2x
4. Evaluate at x = 2: f'(2) = 2(2) = 4

Therefore, the derivative of f(x) = x² at x = 2 is 4. This means the slope of the tangent line to the curve y = x² at x = 2 is 4.

2. Power Rule

The power rule is a significantly more efficient method for finding the derivatives of polynomial functions. If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹.

Example: For f(x) = x³, the derivative is f'(x) = 3x². To find the derivative at x = 2, we substitute: f'(2) = 3(2)² = 12.

3. Sum/Difference Rule

The derivative of a sum or difference of functions is the sum or difference of their derivatives:

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Example: Let f(x) = x³ + 2x - 5. Then f'(x) = 3x² + 2. At x = 1, f'(1) = 3(1)² + 2 = 5.

4. Product Rule

For the product of two functions, u(x) and v(x):

d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

Example: Let f(x) = (x² + 1)(x - 2). Then, using the product rule, f'(x) = 2x(x - 2) + (x² + 1)(1) = 3x² - 4x + 1. At x = 0, f'(0) = 1.

5. Quotient Rule

For the quotient of two functions, u(x) and v(x):

d/dx [u(x)/v(x)] = [u'(x)v(x) - u(x)v'(x)] / [v(x)]²

Example: Let f(x) = (x + 1) / (x - 1). Then, using the quotient rule, f'(x) = [(1)(x - 1) - (x + 1)(1)] / (x - 1)² = -2 / (x - 1)². At x = 2, f'(2) = -2 / (2 - 1)² = -2.

6. Chain Rule

The chain rule is essential for finding derivatives of composite functions (functions within functions). If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Example: Let y = (x² + 1)³. Here, f(u) = u³ and u = g(x) = x² + 1. Then, dy/dx = 3(x² + 1)² * 2x = 6x(x² + 1)². At x = 1, dy/dx = 6(1)(1 + 1)² = 24.

Higher-Order Derivatives

It's also possible to find higher-order derivatives. The second derivative (f''(x)) is the derivative of the first derivative, the third derivative (f'''(x)) is the derivative of the second derivative, and so on. These higher-order derivatives provide information about the function's concavity and other properties.

Applications of Evaluating Derivatives at a Given Point

The ability to evaluate the derivative at a specific point has significant applications in various fields:

• Physics: Finding the instantaneous velocity or acceleration of an object at a particular time.
• Engineering: Determining the rate of change of a system's parameters, such as temperature or pressure.
• Economics: Analyzing marginal cost, marginal revenue, and other economic indicators.
• Machine Learning: Optimizing models using gradient descent algorithms, which rely on derivatives to find the direction of steepest descent.

Handling Different Function Types

The methods described above can be applied to a wide range of functions, including polynomial, rational, trigonometric, exponential, and logarithmic functions. However, some functions may require more advanced techniques, such as implicit differentiation or logarithmic differentiation.

Example (Trigonometric Function):

Let's find the derivative of f(x) = sin(x) at x = π/2.

The derivative of sin(x) is cos(x). Therefore, f'(x) = cos(x). At x = π/2, f'(π/2) = cos(π/2) = 0.

Example (Exponential Function):

Let's find the derivative of f(x) = e^x at x = 0.

The derivative of e^x is e^x. Therefore, f'(x) = e^x. At x = 0, f'(0) = e^0 = 1.

Conclusion

Evaluating the derivative of a function at a given point is a cornerstone of calculus. Understanding the various methods for finding derivatives, from the limit definition to the power, product, quotient, and chain rules, is essential for effectively analyzing the behavior of functions. The applications of this concept span numerous fields, highlighting its importance in understanding rates of change and optimizing systems. While the process may seem complex at first, with consistent practice and a solid grasp of the underlying principles, evaluating derivatives at a specific point becomes a straightforward and powerful tool in mathematical analysis.