Calculus Study Guide: Techniques of Differentiation and Applications

Techniques of Differentiation

Overview of Derivative Rules

This section covers the fundamental rules for computing derivatives, which are essential for analyzing functions and solving calculus problems. Mastery of these rules allows students to tackle a wide range of applications involving rates of change, motion, and tangent lines.

• Power Rule: For any real number n, the derivative of $x^n$ is $nx^{n-1}$.

• Sum Rule: The derivative of a sum is the sum of the derivatives: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$.

• Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$.

• Exponential Rule: The derivative of $e^{x}$ is $e^{x}$, and for $a^{x}$, it is $a^{x} \ln a$.

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

• Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$.

• Trigonometric Functions: For example, $\frac{d}{dx}[\sin x] = \cos x$, $\frac{d}{dx}[\cos x] = -\sin x$, $\frac{d}{dx}[\tan x] = \sec^2 x$, etc.

• Chain Rule: For composite functions, $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$.

• Higher Order Derivatives: The second derivative is the derivative of the first derivative, and so on: $f''(x) = \frac{d^2}{dx^2}f(x)$.

• Combination of Rules: Many functions require using multiple rules together (e.g., product and chain rule).

Example:

• Find the derivative of $y = x^2 \sin(x)$.

  • Apply the product rule: $y' = 2x \sin(x) + x^2 \cos(x)$.

Applications of Derivatives

Rates of Change

Derivatives are used to measure how a quantity changes with respect to another. This is fundamental in physics, engineering, and other sciences.

• Position, Velocity, and Acceleration:

  • Position function: $s(t)$

  • Velocity: $v(t) = s'(t)$

  • Acceleration: $a(t) = v'(t) = s''(t)$

• Units: Always include correct units (e.g., if $s(t)$ is in meters and $t$ in seconds, $v(t)$ is in meters/second).

Example:

• If $s(t) = 3t^2 + 2t$, then $v(t) = 6t + 2$ and $a(t) = 6$.

Tangent Lines

The derivative at a point gives the slope of the tangent line to the curve at that point. The equation of the tangent line to $y = f(x)$ at $x = a$ is:

• $y = f(a) + f'(a)(x - a)$

Example:

• For $f(x) = x^2$ at $x = 1$, $f(1) = 1$, $f'(x) = 2x$, so $f'(1) = 2$. The tangent line is $y = 1 + 2(x - 1)$.

Interpreting Derivatives

Derivatives can be interpreted as instantaneous rates of change. Notation includes $f'(x)$, $\frac{dy}{dx}$, $\frac{d}{dx}f(x)$, etc. Always express answers with correct units and context.

• Example: If $y$ is distance in meters and $x$ is time in seconds, $\frac{dy}{dx}$ is velocity in meters/second.

Best Practices for Studying and Exam Preparation

Study Strategies

• Begin reviewing at least a week before the exam.

• Complete all homework and practice problems from the textbook, worksheets, and online assignments.

• Work through blank versions of notes, quizzes, and worksheets without referring to solutions.

• Write out solutions fully; do not just read through problems.

• Review all objectives and ensure you understand each concept.

• Seek help from instructors, study centers, or peers if needed.

Exam Expectations

• Show all work and reasoning for each problem.

• Provide answers in exact form unless otherwise specified.

• Include correct units in all answers involving physical quantities.

• Be prepared to explain your answers clearly and justify your steps.

Summary Table: Derivative Rules

Rule	Formula	Example
Power Rule	$\frac{d}{dx}[x^n] = nx^{n-1}$	$\frac{d}{dx}[x^3] = 3x^2$
Sum Rule	$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$	$\frac{d}{dx}[x^2 + \sin x] = 2x + \cos x$
Product Rule	$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$	$\frac{d}{dx}[x \sin x] = 1 \cdot \sin x + x \cdot \cos x$
Quotient Rule	$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$	$\frac{d}{dx}\left[\frac{x}{\cos x}\right] = \frac{1 \cdot \cos x - x \cdot (-\sin x)}{(\cos x)^2}$
Chain Rule	$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$	$\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x$
Exponential Rule	$\frac{d}{dx}[e^{x}] = e^{x}$	$\frac{d}{dx}[e^{2x}] = e^{2x} \cdot 2$
Trigonometric Functions	$\frac{d}{dx}[\sin x] = \cos x$	$\frac{d}{dx}[\tan x] = \sec^2 x$

Additional info: This guide is based on the topics listed for Math 131 Test 3, focusing on differentiation techniques and their applications. Students are expected to demonstrate full solutions and clear reasoning on all exam problems.