Calculus I: Derivatives, Related Rates, and Applications Study Guide

Derivatives and Their Applications

Introduction to Derivatives

The derivative is a fundamental concept in calculus, representing the instantaneous rate of change of a function with respect to one of its variables. It is widely used to analyze the behavior of functions, solve problems involving motion, and model real-world phenomena.

• Definition: The derivative of a function f(x) at a point x=a is defined as:

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

• Notation: Common notations include f'(x), \( \frac{df}{dx} \), and Df(x).

• Interpretation: The derivative gives the slope of the tangent line to the graph of the function at a given point.

Related Rates

Solving Related Rates Problems

Related rates problems involve finding the rate at which one quantity changes with respect to time, given information about the rates of change of other related quantities. These problems often require implicit differentiation and the chain rule.

• Key Steps:

  1. Identify all variables that change with time.

  2. Write an equation relating the variables.

  3. Differentiating both sides with respect to time t, using the chain rule as needed.

  4. Substitute known values and solve for the desired rate.

• Example 1: The surface area A of a cylinder is $A = 2\pi r^2 + 2\pi r h$. If the radius r is increasing at a constant rate and h is fixed, find $\frac{dA}{dt}$ when r is a given value.

• Example 2: The volume V of a cone is $V = \frac{1}{3}\pi r^2 h$. If the radius r is increasing at a constant rate and h is fixed, find $\frac{dV}{dt}$ for a given r.

Techniques of Differentiation

Basic Derivative Rules

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

• Constant Multiple Rule: $\frac{d}{dx}(c f(x)) = c f'(x)$

• Sum Rule: $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$

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

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

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

Derivatives of Common Functions

• Exponential Functions: $\frac{d}{dx}(e^{x}) = e^{x}$

• Logarithmic Functions: $\frac{d}{dx}(\ln x) = \frac{1}{x}$

• Trigonometric Functions:

  • $\frac{d}{dx}(\sin x) = \cos x$

  • $\frac{d}{dx}(\cos x) = -\sin x$

  • $\frac{d}{dx}(\tan x) = \sec^2 x$

• Inverse Trigonometric Functions:

  • $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$

  • $\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$

  • $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$

Applications of Derivatives

Motion Along a Line

In physics, the position, velocity, and acceleration of a particle moving along a straight line can be described using derivatives.

• Position Function: $s(t)$ or $x(t)$ gives the location of the particle at time t.

• Velocity: $v(t) = s'(t)$ is the derivative of position with respect to time.

• Speed: $|v(t)|$ is the absolute value of velocity.

• Acceleration: $a(t) = v'(t) = s''(t)$ is the derivative of velocity with respect to time.

• Example: If $s(t) = -11t^2 + 15t - 5$, then $v(t) = \frac{ds}{dt}$ and $a(t) = \frac{d^2s}{dt^2}$.

Implicit Differentiation

Finding Derivatives When y is Defined Implicitly

Implicit differentiation is used when a function is not given explicitly as y = f(x), but rather as an equation involving both x and y.

• Procedure:

  1. Differentiate both sides of the equation with respect to x, treating y as a function of x.

  2. Whenever differentiating a term involving y, multiply by $\frac{dy}{dx}$ (chain rule).

  3. Solve for $\frac{dy}{dx}$.

• Example: For $-y^3 + 5xy + 10x^2 = 1625$, differentiate both sides with respect to x to find $\frac{dy}{dx}$.

Limit Definition of the Derivative

Calculating the Derivative from First Principles

The limit definition provides a foundational approach to finding the derivative at a point, emphasizing the concept of instantaneous rate of change.

• Formula: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Application: Use this definition to find the slope of the tangent line to a function at a specific point, especially when instructed not to use shortcut rules.

• Example: For $g(x) = \frac{10}{7}$ at $x = -4$, apply the limit definition to compute the derivative.

Summary Table: Derivative Rules and Applications

Rule/Concept	Formula	Example
Power Rule	$\frac{d}{dx}(x^n) = n x^{n-1}$	$\frac{d}{dx}(x^3) = 3x^2$
Product Rule	$\frac{d}{dx}(fg) = f'g + fg'$	$\frac{d}{dx}(x^2 \sin x)$
Quotient Rule	$\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f'g - fg'}{g^2}$	$\frac{d}{dx}\left(\frac{x}{x+1}\right)$
Chain Rule	$\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$	$\frac{d}{dx}(\sin(x^2))$
Implicit Differentiation	Differentiate both sides, solve for $\frac{dy}{dx}$	$x^2 + y^2 = 1$
Related Rates	Differentiate with respect to t	Volume of a sphere: $V = \frac{4}{3}\pi r^3$

Additional info:

• Some problems require the use of the limit definition of the derivative rather than shortcut rules.

• Problems include finding derivatives of composite, trigonometric, exponential, and logarithmic functions, as well as applying these to motion and related rates scenarios.