Calculus Exam 2 Study Guide: Derivatives, Chain Rule, Implicit Differentiation, and Related Rates

1. Chain Rule and Derivatives of Composite Functions

1.1 Understanding the Chain Rule

The chain rule is a fundamental technique for differentiating composite functions, where one function is applied inside another. It is essential for finding derivatives when functions are nested.

• Definition: If $y = f(g(x))$, then the derivative is $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

• Application: Used when differentiating functions like $\sin(x^2)$, $e^{3x+1}$, or $\ln(2x^2+1)$.

• Example: If $y = \sin(x^2)$, then $\frac{dy}{dx} = \cos(x^2) \cdot 2x$.

1.2 Derivatives from Graphs

Given graphs of differentiable functions $f(x)$ and $g(x)$, you may be asked to compute derivatives at specific points using tangent lines or provided values.

• Key Point: The slope of the tangent line at a point gives the derivative at that point.

• Example: If the tangent to $f(x)$ at $x=1$ has slope 3, then $f'(1) = 3$.

1.3 Derivatives of Functions Defined by Formulas

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

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

• Example: If $F(x) = \frac{f(x)}{g(f(x))}$, use both chain and quotient rules.

2. Implicit Differentiation

2.1 Concept and Application

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

• Key Point: Differentiate both sides with respect to $x$, treating $y$ as a function of $x$ (so $\frac{d}{dx}y = y'$).

• Example: For $x^2 + y^2 = 1$, $2x + 2y y' = 0 \implies y' = -\frac{x}{y}$.

2.2 Tangent Lines via Implicit Differentiation

• Find the slope: Use implicit differentiation to find $y'$ at a given point.

• Equation of tangent line: $y - y_0 = m(x - x_0)$, where $m$ is the slope at $(x_0, y_0)$.

3. Derivatives as Rates of Change

3.1 Physical Interpretation

The derivative of a function with respect to time often represents a rate of change, such as velocity or acceleration.

• Velocity: $v(t) = s'(t)$, where $s(t)$ is position.

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

• Example: If $s(t) = 2\cos(t) + \sqrt{t}$, then $v(t) = -2\sin(t) + \frac{1}{2\sqrt{t}}$.

3.2 Changing Direction

• Key Point: A particle changes direction when its velocity changes sign.

• To find when this occurs: Solve $v(t) = 0$ for $t$.

4. Related Rates

4.1 Problem Types

Related rates problems involve finding the rate at which one quantity changes by relating it to other quantities whose rates of change are known.

• Common Scenarios: Ladder problems, conical tank, light beam, volume of a sphere, particle moving along a curve.

• General Steps:

  1. Draw a diagram and assign variables.

  2. Write an equation relating the variables.

  3. Differentiate both sides with respect to time $t$.

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

• Example: For a ladder sliding down a wall, relate $x$ (distance from wall) and $y$ (height on wall) using $x^2 + y^2 = L^2$.

5. Logarithmic Differentiation

5.1 Technique

Logarithmic differentiation is useful for differentiating complicated products, quotients, or powers.

• Steps:

  1. Take the natural logarithm of both sides: $\ln y = \ln f(x)$.

  2. Differentiate both sides using implicit differentiation.

  3. Solve for $y'$.

• Example: For $y = (x^2 + 1)^5 \sqrt{x-1} / (1 - x^2)$, take $\ln$ of both sides and differentiate.

6. Limits and the Definition of the Derivative

6.1 Computing Limits

Some limits can be evaluated by relating them to the definition of the derivative:

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

• Example: $\lim_{h \to 0} \frac{\sin(\frac{\pi}{2} + h) - 1}{h}$ is the derivative of $\sin x$ at $x = \frac{\pi}{2}$.

7. Derivatives of Inverse Trigonometric Functions

7.1 Formulas

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

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

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

7.2 Derivation via Implicit Differentiation

• Example: To find $\frac{d}{dx} \arcsin(x)$, set $y = \arcsin(x)$, so $x = \sin(y)$. Differentiate both sides with respect to $x$ and solve for $\frac{dy}{dx}$.

8. Summary Table: Common Derivative Rules

Function	Derivative
$f(x) = x^n$	$f'(x) = n x^{n-1}$
$f(x) = \sin(x)$	$f'(x) = \cos(x)$
$f(x) = \cos(x)$	$f'(x) = -\sin(x)$
$f(x) = e^x$	$f'(x) = e^x$
$f(x) = \ln(x)$	$f'(x) = \frac{1}{x}$
$f(x) = \arcsin(x)$	$f'(x) = \frac{1}{\sqrt{1-x^2}}$
$f(x) = \arctan(x)$	$f'(x) = \frac{1}{1+x^2}$

9. Additional Info

• Be prepared for problems involving the computation of derivatives from graphs, tables, and formulas.

• Practice implicit differentiation and related rates, as these are commonly tested.

• Understand the geometric and physical interpretations of the derivative, especially as a rate of change.

• Know how to use the definition of the derivative to compute limits.

• Be able to derive and use the formulas for the derivatives of inverse trigonometric functions.