Definition of a Derivative (Rate of Change)

Introduction

Key Concepts

1. What is a Derivative?

A derivative measures how a function changes as its input changes. Formally, the derivative of a function \( f \) at a point \( x \) is the limit: $$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$ This limit, if it exists, represents the slope of the tangent line to the graph of \( f \) at the point \( x \). In simpler terms, it quantifies the instantaneous rate of change of the function with respect to its variable.

2. Geometric Interpretation

Geometrically, the derivative at a point corresponds to the slope of the function's graph at that specific point. If you imagine zooming in on the graph near \( x \), the curve approaches a straight line—the tangent—which has a slope equal to the derivative \( f'(x) \).

3. Physical Interpretation

In physics, the derivative has a direct interpretation as a rate of change. For instance, if \( s(t) \) represents the position of an object over time, then \( s'(t) \) is the velocity, indicating how position changes with time.

4. Differentiability

A function is said to be differentiable at a point \( x \) if the derivative \( f'(x) \) exists at that point. Differentiability implies that the function is smooth (i.e., has no sharp corners or cusps) at \( x \). Not all functions are differentiable everywhere; points where the derivative does not exist are called non-differentiable points.

5. Rules of Differentiation

Calculating derivatives efficiently involves utilizing various differentiation rules:

• Power Rule: For any real number \( n \), \( \frac{d}{dx} x^n = n x^{n-1} \).
• Product Rule: If \( u(x) \) and \( v(x) \) are differentiable, then \( \frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \).
• Quotient Rule: If \( u(x) \) and \( v(x) \) are differentiable, then \( \frac{d}{dx} \left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \).
• Chain Rule: For composite functions \( f(g(x)) \), the derivative is \( f'(g(x)) \cdot g'(x) \).

6. Higher-Order Derivatives

While the first derivative represents the rate of change, higher-order derivatives offer deeper insights. The second derivative, \( f''(x) \), indicates the concavity of the function and is associated with acceleration in physical contexts. Higher derivatives can provide information about the function's behavior and its changes in concavity or inflection points.

7. Implicit Differentiation

Not all functions are given in the form \( y = f(x) \). When dealing with equations where \( y \) is defined implicitly in terms of \( x \), implicit differentiation is employed. By differentiating both sides of the equation with respect to \( x \) and solving for \( \frac{dy}{dx} \), we can find the derivative even when \( y \) is not explicitly isolated.

8. Applications of Derivatives

Derivatives have a wide range of applications across various fields:

• Optimization: Finding maximum and minimum values of functions, crucial in economics and engineering.
• Motion Analysis: Determining velocity and acceleration from position functions.
• Curve Sketching: Analyzing the shape and behavior of graphs by examining derivatives.
• Related Rates: Solving problems where multiple quantities change with respect to time.

9. Differentiability vs. Continuity

While differentiability implies continuity, the converse is not true. A function might be continuous at a point but not differentiable there. For example, \( f(x) = |x| \) is continuous everywhere but not differentiable at \( x = 0 \) due to the sharp corner.

10. Differentiable Functions

Common differentiable functions include polynomial functions, exponential functions, logarithmic functions, and trigonometric functions. These functions are smooth and have well-defined derivatives across their domains, making them fundamental in calculus.

11. Notation of Derivatives

Derivatives can be expressed in various notations:

• Leibniz's Notation: \( \frac{dy}{dx} \).
• Lagrange's Notation: \( f'(x) \).
• Newton's Notation: \( \dot{y} \) (primarily used in physics for time derivatives).

12. Differentiation of Parametric and Polar Functions

Derivatives can also be found for parametric and polar equations. For parametric equations \( x(t) \) and \( y(t) \), the derivative \( \frac{dy}{dx} \) is given by: $$ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} $$ Similarly, for polar coordinates \( r(\theta) \), the derivative expresses the rate of change of the radius with respect to the angle.

13. Linear Approximation

The derivative provides a means to approximate functions linearly near a specific point. The linear approximation of \( f \) at \( x = a \) is: $$ L(x) = f(a) + f'(a)(x - a) $$ This approximation is useful for estimating function values and understanding local behavior.

14. Differentials

Differentials extend the concept of derivatives by considering infinitesimal changes. If \( dy = f'(x)dx \), where \( dx \) is an infinitesimal change in \( x \), then \( dy \) represents the corresponding change in \( y \). Differentials are instrumental in applications such as error estimation and integral approximations.

15. Techniques for Finding Derivatives

Beyond the basic rules, several advanced techniques aid in finding derivatives:

• Logarithmic Differentiation: Useful for functions with complex exponents or products.
• Implicit Differentiation: Necessary when dealing with implicitly defined functions.
• Differentiation Under the Integral Sign: Applies to integrals dependent on parameters.

Comparison Table

Summary and Key Takeaways