Rules of Differentiation: Fundamental Techniques and Applications 3.3

Rules of Differentiation

Introduction

Differentiation is a central concept in calculus, allowing us to compute the rate of change of functions. The following rules provide systematic methods for finding derivatives of various types of functions, which are essential for solving problems in mathematics, physics, engineering, and other fields.

Constant Rule

The derivative of a constant function is always zero, reflecting that constants do not change.

• Rule: If $f(x) = c$, then $f'(x) = 0$.

• Example: $\frac{d}{dx}(28) = 0$

Power Rule

The power rule is used to differentiate functions of the form $x^n$, where $n$ is any real number.

• Rule: $\frac{d}{dx}(x^n) = n x^{n-1}$

• Example: $\frac{d}{dx}(x^9) = 9x^8$

Constant Multiple Rule

This rule states that the derivative of a constant times a function is the constant times the derivative of the function.

• Rule: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$

• Example: $\frac{d}{dx}(-5x^{20}) = -5 \cdot 20x^{19} = -100x^{19}$

Sum and Difference Rule

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

• Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

• Example: $\frac{d}{dx}(12x^5 - 40x + e^x - e^4) = 60x^4 - 40 + e^x$

Derivative of the Exponential Function

The exponential function $e^x$ is unique in that its derivative is itself.

• Rule: $\frac{d}{dx}(e^x) = e^x$

• Example: $\frac{d}{dx}(e^x) = e^x$

Examples of Differentiation

• Example 1: $f(x) = 6 \sqrt{x} = 6x^{1/2}$ $f'(x) = 6 \cdot \frac{1}{2} x^{-1/2} = 3x^{-1/2}$

• Example 2: $y = 5x^{-3} + \frac{4}{x} = 5x^{-3} + 4x^{-1}$ $y' = 5 \cdot (-3)x^{-4} + 4 \cdot (-1)x^{-2} = -15x^{-4} - 4x^{-2}$

Applications: Tangent Lines and Horizontal Tangents

Derivatives are used to find the slope of tangent lines and points where the tangent is horizontal.

• Finding the slope at a point: For $f(x) = 6x^2 - 8x$, find $x$ where the slope is $16$:$f'(x) = 12x - 8$Set $12x - 8 = 16 \Rightarrow x = 2$ $f(2) = 8$ Point: $(2, 8)$

• Finding horizontal tangents: For $f(x) = x^3 + 4x^2 - 3x$, set $f'(x) = 0$: $f'(x) = 3x^2 + 8x - 3$ $3x^2 + 8x - 3 = 0$ Solutions: $x = -3$, $x = \frac{1}{3}$

Equation of the Tangent Line

The equation of the tangent line to $f(x)$ at $x=a$ is given by $y - f(a) = f'(a)(x - a)$.

• Example: For $f(x) = x^3 + 4x^2 - 3x$ at $(1, 2)$: $f'(x) = 3x^2 + 8x - 3$ $f'(1) = 8$ Equation: $y - 2 = 8(x - 1) \Rightarrow y = 8x - 6$

Higher Order Derivatives

Second and higher order derivatives measure the rate of change of the rate of change, useful for analyzing concavity and acceleration.

• Second derivative: $f''(x) = (f'(x))'$ $\frac{d^2 f}{dx^2}$ or $\frac{d^2 y}{dx^2}$

• nth derivative: $f^{(n)}(x) = (f^{(n-1)}(x))'$ $y^{(n)}$, $\frac{d^n f}{dx^n}$, $\frac{d^n y}{dx^n}$

Example: First and Second Derivatives

• Example: $f(x) = 2x^3 + 9x^2 + 6x$ $f'(x) = 6x^2 + 18x + 6$ $f''(x) = 12x + 18$