Calculus in a Nutshell

• Definition: The rate of change [slope] of a function at a point is the limiting value of its average slope over an interval including that point, as the width of the interval shrinks to zero:
  
  $${dy \over dx} \; \equiv \; \lim_{\Delta x \to 0} \; {y(x + \Delta x) \; - \; y(x) \over \Delta x }$$
  
  
  All the remaining Laws and Rules can be proven by algebraic manipulation of this definition.

• Operator Notation: The symbol $${d \over dx}$$ (read "derivative with respect to x") can be thought of as a mathematical "verb" (called an operator) which "operates on" whatever we place to its right. Thus
  
  $${d \over dx} \left[ y \right ] \; \equiv \; {dy \over dx}$$
  
  

• Power Law: The simplest class of derivatives are those of power-law functions:
  
  $${d \over dx} \left[ x^p \right] \; = \; p \, x^{p-1}$$
  
  
  valid for all powers p, whether positive, negative, integer, rational, irrational, real, imaginary or complex.

• Product Law: The derivative of the product of two functions is not the product of their derivatives! Instead,
  
  $${d \over dx} \left[ f(x) \cdot g(x) \right] \; = \; {df \over dx} \cdot g(x) \; + \; f(x) \cdot {dg \over dx}$$
  
  

• Constant times a Function: The Product Law gives
  
  $${d \over dx} \left[ a \cdot y(x) \right] \; = \; a \cdot {dy \over dx}$$
  
  
  where a is a constant (i.e., not a function of x). This is often referred to as "pulling the constant factor outside the derivative."

• Function of a Function: Suppose y is a function of x and x is in turn a function of t.
  
  $$\qquad \qquad \hbox{\rm Then} \qquad \qquad {d \over dt} \l . . . . . . t {dx \over dt} \qquad \qquad \hbox{\rm ({\bf Chain Rule})}.$$
  
  

• Exponentials:
  
  $${d \over dx} \left[ e^{k x} \right] \; = \; k \cdot e^{k x}$$
  
  
  where k is any constant.

• Natural Logarithms:
  
  $${d \over dx} \left[ \ln x \right] \; = \; {1 \over x}$$
  
  

• Antiderivatives: We can "solve" integrals or "antiderivatives" the same way we "solve" long division problems: by trial and error guessing!  Suppose we are given an explicit function f(x) [for example, f(x) = x²] and told that f(x) is the derivative of a function y(x) which we would like to know - that is,
  
  $$\hbox{\rm If} \qquad {dy \over dx} = x^2, \qquad \hbox{\rm what is} \qquad y(x) \, ?$$
  
  
  Well, we know that
  
  $${d \over dx} \left[ x^3 \right] \; = \; 3 \, x^2,$$
  
  
  so we must divide by 3 to get
  
  $$y(x) = {\textstyle {1 \over 3}} \, x^3 \; + \; y_0$$
  
  
  where the constant term y₀ (the value of y when x=0) cannot be determined from the information given - the derivative of any constant is zero, so such an integral is always undetermined to within such a constant of integration.