WhizMath

Introduction: Understanding Rates of Change

Welcome to Whizmath, where we delve into the dynamic world of calculus! Today's lesson introduces one of the most fundamental concepts in mathematics: Derivatives. At its core, a derivative measures how a function changes as its input changes. It's the mathematical tool for understanding rates of change, slopes of curves, and instantaneous velocities.

From predicting the trajectory of a rocket and optimizing production costs in business, to analyzing the speed of chemical reactions and understanding the flow of electricity, derivatives are indispensable across physics, engineering, economics, and countless scientific disciplines. They allow us to move beyond average rates of change to pinpoint the exact rate of change at any given moment.

In this comprehensive lesson, we will define derivatives using limits, explore various rules for differentiation, interpret derivatives graphically, and apply them to real-world problems. Prepare to unlock the power of instantaneous change and gain a deeper insight into how quantities evolve!

Chapter 1: The Concept of a Derivative

To truly understand derivatives, we must first grasp the idea of a limit and how it relates to the slope of a tangent line.

1.1 Average Rate of Change vs. Instantaneous Rate of Change

• Average Rate of Change: The change in a function's output over a specific interval. For a function $f(x)$ between $x_1$ and $x_2$, it's given by the slope of the secant line:

  $\frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• Instantaneous Rate of Change: The rate of change at a single, specific point. This is where the derivative comes in. It's the slope of the tangent line to the curve at that point.

1.2 Definition of the Derivative (Limit Definition)

The derivative of a function $f(x)$ with respect to $x$, denoted as $f'(x)$ (read "f prime of x") or $\frac{dy}{dx}$, is defined as:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

This limit represents the slope of the tangent line to the graph of $f(x)$ at any point $x$.

Example 1.2.1: Using the Limit Definition

Chapter 2: Basic Differentiation Rules

While the limit definition is fundamental, using it for every function is tedious. Fortunately, we have powerful rules to simplify differentiation.

2.1 Constant Rule

The derivative of a constant function is zero.

$\frac{d}{dx}(c) = 0$

Example: $\frac{d}{dx}(5) = 0$

2.2 Power Rule

If $n$ is any real number, the derivative of $x^n$ is $nx^{n-1}$.

$\frac{d}{dx}(x^n) = nx^{n-1}$

Example: $\frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2$

Example: $\frac{d}{dx}(\sqrt{x}) = \frac{d}{dx}(x^{1/2}) = \frac{1}{2}x^{1/2 - 1} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$

2.3 Constant Multiple Rule

The derivative of a constant times a function is the constant times the derivative of the function.

$\frac{d}{dx}(cf(x)) = c \frac{d}{dx}(f(x))$

Example: $\frac{d}{dx}(5x^4) = 5 \frac{d}{dx}(x^4) = 5(4x^3) = 20x^3$

2.4 Sum and Difference Rules

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

$\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x))$

Example: $\frac{d}{dx}(3x^2 + 2x - 7) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(2x) - \frac{d}{dx}(7) = 6x + 2 - 0 = 6x + 2$

2.5 Product Rule

If $f(x)$ and $g(x)$ are differentiable functions, then the derivative of their product is:

$\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$

Example: Find $\frac{d}{dx}((x^2+1)(x^3-2))$

Let $f(x) = x^2+1 \Rightarrow f'(x) = 2x$

Let $g(x) = x^3-2 \Rightarrow g'(x) = 3x^2$

$\frac{d}{dx}((x^2+1)(x^3-2)) = (2x)(x^3-2) + (x^2+1)(3x^2)$

$ = 2x^4 - 4x + 3x^4 + 3x^2$

$ = 5x^4 + 3x^2 - 4x$

2.6 Quotient Rule

If $f(x)$ and $g(x)$ are differentiable functions and $g(x) \neq 0$, then the derivative of their quotient is:

$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$

Example: Find $\frac{d}{dx}\left(\frac{x^2}{x+1}\right)$

Let $f(x) = x^2 \Rightarrow f'(x) = 2x$

Let $g(x) = x+1 \Rightarrow g'(x) = 1$

$\frac{d}{dx}\left(\frac{x^2}{x+1}\right) = \frac{(2x)(x+1) - (x^2)(1)}{(x+1)^2}$

$ = \frac{2x^2 + 2x - x^2}{(x+1)^2}$

$ = \frac{x^2 + 2x}{(x+1)^2}$

2.7 Chain Rule

Used for differentiating composite functions (a function within a function). If $y = f(g(x))$, then:

$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

Example: Find $\frac{d}{dx}((2x+3)^5)$

Let $f(u) = u^5$ and $u = g(x) = 2x+3$.

$f'(u) = 5u^4$ and $g'(x) = 2$

$\frac{d}{dx}((2x+3)^5) = 5(2x+3)^4 \cdot 2 = 10(2x+3)^4$

Chapter 3: Derivatives of Common Functions

Beyond polynomial functions, derivatives of trigonometric, exponential, and logarithmic functions are essential.

3.1 Trigonometric Functions

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

Example: $\frac{d}{dx}(\sin(3x)) = \cos(3x) \cdot \frac{d}{dx}(3x) = 3\cos(3x)$ (using Chain Rule)

3.2 Exponential and Logarithmic Functions

• $\frac{d}{dx}(e^x) = e^x$
• $\frac{d}{dx}(a^x) = a^x \ln a$
• $\frac{d}{dx}(\ln x) = \frac{1}{x}$
• $\frac{d}{dx}(\log_b x) = \frac{1}{x \ln b}$

Example: $\frac{d}{dx}(e^{x^2}) = e^{x^2} \cdot \frac{d}{dx}(x^2) = 2xe^{x^2}$ (using Chain Rule)

Example: $\frac{d}{dx}(\ln(5x)) = \frac{1}{5x} \cdot \frac{d}{dx}(5x) = \frac{1}{5x} \cdot 5 = \frac{1}{x}$

Chapter 4: Applications of Derivatives

Derivatives are powerful tools for solving a wide range of problems involving change and optimization.

4.1 Slope of a Tangent Line

The derivative $f'(x)$ gives the slope of the line tangent to the curve $y=f(x)$ at any point $x$.

Example 4.1.1:

Find the slope of the tangent line to $f(x) = x^3 - 2x$ at $x=2$.

4.2 Rates of Change (Physics/Kinematics)

• If $s(t)$ is the position function of an object, then:
  • Velocity: $v(t) = s'(t) = \frac{ds}{dt}$
  • Acceleration: $a(t) = v'(t) = s''(t) = \frac{d^2s}{dt^2}$

Example 4.2.1:

A ball's height (in meters) is given by $s(t) = -5t^2 + 20t + 1$. Find its velocity at $t=1$ second.

4.3 Optimization (Finding Maxima and Minima)

Derivatives are used to find local maximum and minimum values of a function. These occur at critical points, where $f'(x) = 0$ or $f'(x)$ is undefined.

Steps for Optimization:

1. Find the first derivative $f'(x)$.
2. Set $f'(x) = 0$ and solve for $x$ to find critical points.
3. Use the First Derivative Test (sign changes of $f'(x)$) or Second Derivative Test ($f''(x)$) to classify critical points as local maxima, minima, or neither.

Example 4.3.1:

Find the local extrema of $f(x) = x^3 - 3x^2 + 2$.

Chapter 5: Higher-Order Derivatives and Implicit Differentiation

Derivatives can be taken multiple times, and we can differentiate equations where $y$ is not explicitly defined as a function of $x$.

5.1 Higher-Order Derivatives

The second derivative, third derivative, and so on are called higher-order derivatives.

• Second Derivative: The derivative of the first derivative, denoted $f''(x)$ or $\frac{d^2y}{dx^2}$. It measures the rate of change of the slope (concavity).
• Third Derivative: The derivative of the second derivative, $f'''(x)$ or $\frac{d^3y}{dx^3}$.

Example: If $f(x) = x^4 - 3x^2 + 5x$, find $f''(x)$.

$f'(x) = 4x^3 - 6x + 5$

$f''(x) = \frac{d}{dx}(4x^3 - 6x + 5) = 12x^2 - 6$

5.2 Implicit Differentiation

Used when an equation defines $y$ implicitly as a function of $x$ (e.g., $x^2 + y^2 = 25$). Differentiate both sides of the equation with respect to $x$, remembering to apply the chain rule when differentiating terms involving $y$ (since $y$ is a function of $x$, $\frac{d}{dx}(y^n) = ny^{n-1} \frac{dy}{dx}$).

Example 5.2.1:

Find $\frac{dy}{dx}$ for $x^2 + y^2 = 25$.

Chapter 6: Common Pitfalls and Tips for Success

Derivatives can be challenging, but being aware of common mistakes can significantly improve your accuracy:

• Algebra Errors: Many "calculus" errors are actually algebra mistakes (e.g., simplifying expressions, distributing negatives).
• Forgetting the Chain Rule: This is perhaps the most common mistake. Always apply the chain rule when differentiating a composite function.
• Misapplying Product/Quotient Rules: Ensure you correctly identify $f(x), g(x), f'(x), g'(x)$ and follow the formula precisely.
• Negative Exponents/Radicals: Convert radicals to fractional exponents and handle negative exponents correctly before applying the power rule.
• Implicit Differentiation: Remember to multiply by $\frac{dy}{dx}$ (or $\frac{d(\text{variable})}{d(\text{independent variable})}$) whenever differentiating a term involving the dependent variable.

Tips for Success:

• Master the Basic Rules: Power, Constant Multiple, Sum/Difference rules are foundational.
• Practice the Chain Rule Extensively: It's the most versatile and frequently used rule.
• Simplify First: Sometimes, algebraic simplification of the original function can make differentiation easier.
• Show Your Work: Write out each step, especially when using product, quotient, or chain rules. This helps in debugging.
• Understand the Concepts: Don't just memorize formulas. Understand what a derivative represents (slope, rate of change).

Practice Problems (with Solutions)