Calculus Study Guide: Derivatives, Differentiability, and Applications

Derivatives and Their Properties

Critical Points and Zero Derivatives

Critical points of a function occur where its derivative is zero or undefined. These points are important in identifying local maxima, minima, and points of inflection.

• Definition: The derivative of a function at a point measures the instantaneous rate of change (slope) of the function at that point.

• Critical Points: If $f'(x) = 0$, then $x$ is a critical point of $f(x)$.

• Example: Given a graph of $f(x)$, the $x$-values where the tangent is horizontal (slope zero) are the solutions to $f'(x) = 0$.

Units of Derivatives

When interpreting the derivative of a function that models a physical situation, the units of the derivative are determined by the units of the original function and its variable.

• Key Point: If $f(t)$ gives position in feet and $t$ is in seconds, then $f'(t)$ has units of feet per second.

• Application: In motion problems, the derivative of position with respect to time is velocity.

Derivative Rules: Product, Quotient, and Chain Rule

Different rules are used to compute derivatives of composite, product, and quotient functions.

• Product Rule: Used when differentiating the product of two functions: $(uv)' = u'v + uv'$

• Quotient Rule: Used for the quotient of two functions: $rac{d}{dx}igg( rac{u}{v}igg) = rac{u'v - uv'}{v^2}$

• Chain Rule: Used for composite functions: $rac{d}{dx}f(g(x)) = f'(g(x)) imes g'(x)$

• Example: For $f(x) = rac{ an(2x+1)}{x^2+3}$, the quotient rule and chain rule are required, but not the product rule.

Differentiability and Graphical Analysis

Points of Non-Differentiability

A function is not differentiable at points where it is not continuous, has a sharp corner, or a vertical tangent.

• Key Point: On a graph, look for jumps, cusps, or vertical tangents to identify where $f(x)$ is not differentiable.

• Example: For a given graph, state all $x$ in the interval $(-6,4)$ where $f$ is not differentiable.

Piecewise Functions and Their Derivatives

Piecewise functions may have points where the derivative does not exist due to discontinuities or sharp turns.

• Key Point: At the boundaries between pieces, check for continuity and matching slopes.

• Example: The derivative of a piecewise linear function is constant on each interval, but may be undefined at the endpoints.

Using Tables to Compute Derivatives

Tabular Data for Functions and Their Derivatives

Tables can provide values of functions and their derivatives at specific points, which can be used to compute derivatives of composite or combined functions.

x	f(x)	g(x)	f'(x)	g'(x)
3	25	4	6	2

• Example: If $h(x) = g(f(x))$, then $h'(x) = g'(f(x)) imes f'(x)$.

• Application: Use the table to find $h'(3)$ for various definitions of $h(x)$.

Average and Instantaneous Rates of Change

Average Velocity

The average rate of change of a function over an interval is given by the difference quotient.

• Formula: $ext{Average velocity} = rac{s(t_2) - s(t_1)}{t_2 - t_1}$

• Example: For $s(t) = 2t^2 - 3t + 1$, compute the average velocity from $t=2$ to $t=2.6$.

Instantaneous Velocity

The instantaneous rate of change at a point is the derivative at that point.

• Formula: $v(t) = s'(t)$

• Example: Find $s'(2)$ for $s(t) = 2t^2 - 3t + 1$.

Limit Definition of the Derivative

Formal Definition

The derivative of a function at a point is defined as the limit of the difference quotient as the interval approaches zero.

• Formula: $f'(x) = rac{d}{dx}f(x) = ext{lim}_{h o 0} rac{f(x+h) - f(x)}{h}$

• Application: Use this definition to find the derivative of $f(x) = rac{1}{x^5}$.

Computing Derivatives: Practice Problems

Basic Derivatives

• Power Rule: $rac{d}{dx}x^n = nx^{n-1}$

• Example: $rac{d}{dx}8x^{-2} = -16x^{-3}$

Sum and Difference Rule

• Rule: $rac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$

• Example: $rac{d}{dx}(x^2 + 5x + 4) = 2x + 5$

Product and Chain Rule Applications

• Product Rule: $rac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

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

• Example: $rac{d}{dx}(x^2 ext{sin}(3x) + ext{sec}(x))$

Exponential and Logarithmic Derivatives

• Exponential: $rac{d}{dx}e^{ax} = a e^{ax}$

• Logarithmic: $rac{d}{dx} ext{ln}(x) = rac{1}{x}$

• Example: $rac{d}{dx} ext{ln}(x^2 + 5) = rac{2x}{x^2 + 5}$

Second Derivatives

• Definition: The second derivative $f''(x)$ measures the rate of change of the rate of change (concavity).

• Example: If $f(x) = ext{sqrt}(x^2 + 4)$, find $f''(5)$.

Summary Table: Derivative Rules

Function Type	Rule	Formula
Power	Power Rule	$rac{d}{dx}x^n = nx^{n-1}$
Product	Product Rule	$rac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$
Quotient	Quotient Rule	$rac{d}{dx}igg( rac{u}{v}igg) = rac{u'v - uv'}{v^2}$
Chain	Chain Rule	$rac{d}{dx}f(g(x)) = f'(g(x))g'(x)$
Exponential	Exponential Rule	$rac{d}{dx}e^{ax} = a e^{ax}$
Logarithmic	Logarithmic Rule	$rac{d}{dx} ext{ln}(x) = rac{1}{x}$

Additional info:

• Some questions require interpreting graphs to determine differentiability and critical points.

• Tabular data is used for composite function derivatives, emphasizing the chain rule.

• Practice problems cover a range of derivative rules, including power, product, quotient, chain, exponential, and logarithmic functions.