Contents
Review
Definition of the Derivative
f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h)  f(x)}{h}
The derivative at a point is the slope of the tangent line to the graph of the function at that point.
Derivative Formulas
Constant Rule: f(x) = c \quad f'(x) = 0
Constant Multiplier Rule: f(x) = c \cdot g(x) \quad f'(x) = c \cdot g'(x)
Power Rule: f(x) = x^n \quad f'(x) = nx^{n1}
Sum Rule: f(x) = g(x) + h(x) \quad f'(x) = g'(x) + h'(x)
Product Rule: f(x) = g(x) \cdot h(x) \quad f'(x) = g(x) \cdot h'(x) + g'(x) \cdot h(x)
Quotient Rule: f(x) = \frac{g(x)}{h(x)} \quad f'(x) = \frac{h(x) g'(x)  h'(x) g(x)}{[h(x)]^2}
Chain Rule: f(x) = g \circ h (x) \quad f'(x) = g' \circ h (x) h'(x)
Differential Equations
We will be dealing with linear differential equations with constant coefficients only.
Homogeneous Solution
To find the homogeneous solution to a differential equation, find the roots of the characteristic polynomial. Recall that the characteristic polynomial is formed by sending y^{(n)} to r^n where y^{(n)} denotes the n^{th} derivative of y and y^{(0)} = y. Then use the table below to form the homogeneous solution, y_h, for your differential equation:
Roots 
Solutions 
one root: r 
y_h = Ae^{rt} 
two distinct roots: r1, r2 
y_h = Ae^{r_1 t} + Be^{r_2 t} 
one repeated root: r 
y_h = Ae^{r t} + Bte^{r t} 
two complex roots: a \pm bi 
y_h = Ae^{a t} \cos(bt) + Be^{a t} \sin (b t) 
Particular Solution
To find a particular solution to a differential equation, make a guess with undetermined coefficients based on the righthand side of your equation. Then plug the guess back into your differential equation and determine the coefficients. The following table for guesses is helpful:
Righthand Side 
Guess 
a_n t^n + ... + a_1 t + a_0 (a_n \neq 0) 
b_n t^n + ... + b_1 t + b_0 
Ae^{at} 
Be^{at} 
C \sin(at) + D \cos(at) 
A \sin(at) + B \cos(at) 
Remember that if your guess "doesn't work", i.e. it is a homogeneous solution (looks like y_h), then you multiply your guess by t.
General Solution
The general solution is the sum of the homogeneous and particular solutions: y = y_h + y_p
Initial Values
If initial values are given, then use them to determine the coefficients in your general solution.
Practice Exercises
Using the derivative formulas above, compute each of the following derivatives in questions 110. You may replace the letters serving as constants with your favorite numbers (a,b,c,d,e, and f) first if you wish. In particular, note that making some constants equal to one will greatly simplify the problem.
y(x) = ax^b
y(x) = a(b + cx^d)^e
y(x) = \frac{ax^b + c}{dx^e + f}
y(x) = a
Past Quizzes
Quiz 
Solutions 
LaTeX source 
 
 

Questions
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