Derivatives

The key word is CHANGE!

Consider a function y(x) of the variable x.  A plot of an arbitrary function y(x) versus x is shown below.  We can find the change in y, ∆y, given a change in x, ∆x.  ∆y/∆x defines the slope of a straight line connecting the points A and B in the figure below.  As B moves closer to A, this straight line approaches the line tangent to y(x) at point A.


image

The slope of this tangent line is the derivative of y with respect to x and is denoted dy/dx.  Since a tangent line to the function y(x) can be defined at any point x, the derivative itself is a function of x.  We can denote it by g(x).

g(x) = dy/dx

The slope of the tangent line at some point on the function y(x) may be approximated by the slope of a line connecting two points, A and B, set a finite distance apart on the curve.

dy/dx ~ ∆y/∆x

The smaller we make the distance ∆x, the better the approximation becomes.  In the limit ∆x --> 0, i.e. when the distance between A and B becomes infinitesimally small, the approximation becomes exact.

Link:  Exploring derivatives using a spreadsheet


Since the tangent line to the function y(x) can be defined at any point x, the derivative itself is a function of x.  The table below lists derivatives of some common functions.  In all expressions a and b are constants.

y(x) = dy/dx =
a*xb a*b*xb-1
a*exp(bx) a*b*exp(bx)
a*sin(bx) a*b*cos(bx)
a*cos(bx) -a*b*sin(bx)
a*ln(x) a/x

Often we will encounter functions of the form y(x) = a*exp(bx + c), y(x) = a*sin(bx + c), and y(x) = a*cos(bx + c).

The derivatives of those functions are given below.

y(x) = dy/dx =
a*exp(bx + c) a*b*exp(bx + c)
a*sin(bx + c) a*b*cos(bx + c)
a*cos(bx + c) -a*b*sin(bx + c)