Defining the Derivative of a Function and Using Derivative Notation

Introduction

Key Concepts

1. Definition of the Derivative

The derivative of a function measures how the function's output value changes as its input changes. Formally, for a function \( f(x) \), the derivative at a point \( x \) is defined as:

This limit, if it exists, represents the slope of the tangent line to the graph of the function at the point \( x \). It provides the instantaneous rate of change of the function with respect to its variable.

2. Derivative Notation

Derivatives can be denoted in several ways, each serving specific purposes in different contexts:

• Lagrange's Notation: \( f'(x) \) signifies the first derivative of \( f \) with respect to \( x \).
• Leibniz's Notation: \( \frac{dy}{dx} \) is used when dealing with functions \( y = f(x) \), emphasizing the variable with respect to which differentiation is performed.
• Newton's Notation: \( \dot{f}(x) \) is commonly used in physics, particularly in mechanics, to denote derivatives with respect to time.

3. Interpretation of the Derivative

Beyond its formal definition, the derivative has various interpretations:

• Graphical Interpretation: The derivative at a point corresponds to the slope of the tangent line to the function's graph at that point.
• Physical Interpretation: In motion, the derivative of the position function with respect to time is velocity, and the derivative of velocity is acceleration.
• Economic Interpretation: Derivatives can represent marginal cost or marginal revenue, indicating how these quantities change with production levels.

4. Rules of Differentiation

Several fundamental rules facilitate the computation of derivatives:

• Power Rule: For any real number \( n \), \( \frac{d}{dx}x^n = nx^{n-1} \).
• Product Rule: \( \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \).
• Quotient Rule: \( \frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2} \).
• Chain Rule: \( \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \).

5. Higher-Order Derivatives

Higher-order derivatives represent the derivatives of derivatives. The second derivative, \( f''(x) \), describes the concavity of the function, while the third derivative, \( f'''(x) \), can provide insights into the rate of change of concavity, and so on.

6. Differentiability and Continuity

For a function to be differentiable at a point, it must be continuous there. However, continuity alone does not guarantee differentiability. Sharp corners or cusps in the graph of a function indicate points where the function is not differentiable.

7. Applications of Derivatives

Derivatives have a wide range of applications, including:

• Optimization: Finding maximum and minimum values of functions, crucial in fields like economics and engineering.
• Motion Analysis: Determining velocity and acceleration from position functions.
• Curve Sketching: Analyzing the behavior of graphs based on derivative information.
• Related Rates: Solving problems involving rates at which related variables change.

8. Implicit Differentiation

When functions are defined implicitly rather than explicitly, implicit differentiation is employed. For example, to differentiate \( x^2 + y^2 = r^2 \) with respect to \( x \), treat \( y \) as a function of \( x \) and apply differentiation accordingly:

9. Derivatives of Trigonometric Functions

The derivatives of basic trigonometric functions are crucial for solving various calculus problems:

• \( \frac{d}{dx} \sin(x) = \cos(x) \)
• \( \frac{d}{dx} \cos(x) = -\sin(x) \)
• \( \frac{d}{dx} \tan(x) = \sec^2(x) \)

10. Exponential and Logarithmic Functions

Differentiating exponential and logarithmic functions involves specific rules:

• \( \frac{d}{dx} e^x = e^x \)
• \( \frac{d}{dx} a^x = a^x \ln(a) \) where \( a > 0 \) and \( a \neq 1 \)
• \( \frac{d}{dx} \ln(x) = \frac{1}{x} \)
• \( \frac{d}{dx} \log_a(x) = \frac{1}{x \ln(a)} \)

11. Differentiation of Inverse Functions

If \( y = f^{-1}(x) \) is the inverse of \( y = f(x) \), then:

This formula is particularly useful when dealing with inverse trigonometric functions.

12. Differentiation Techniques for Complex Functions

For more intricate functions, combinations of differentiation rules are applied. For instance, differentiating \( f(x) = (3x^2 + 2x)(\sin(x)) \) requires the product rule:

13. Implications of the Mean Value Theorem

The Mean Value Theorem states that if a function \( f \) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \( (a, b) \), then there exists at least one point \( c \in (a, b) \) such that:

This theorem has significant implications in proving various other results in calculus.

14. Taylor and Maclaurin Series

Derivatives are essential in constructing Taylor and Maclaurin series, which approximate functions as infinite sums of their derivatives at a specific point. For example, the Taylor series of \( f(x) \) around \( a \) is:

15. Differentiating Parametric and Polar Equations

When functions are defined parametrically or in polar coordinates, differentiation techniques adapt accordingly. For parametric equations \( x(t) \) and \( y(t) \), the derivative \( \frac{dy}{dx} \) is:

For polar equations \( r(\theta) \), differentiation involves converting to Cartesian coordinates or applying specific polar differentiation formulas.

Comparison Table

Summary and Key Takeaways