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Derivatives
© Copyright 2000, Jim Loy
We can easily determine the slope of a straight line, the rise
over the run, a change in y over a change in x. How do we find the slope of a
curve? Before I answer that, what could we possibly mean by the slope of a
curve? Well, a curve has a steepness, just like a line. But, unlike a line, a
curve's steepness varies. So, we will have to look at the steepness (slope) at
a point. In the diagram, we approximate the slope at point (x,y) with a
straight line tangent to the curve. This is a line that just touches the curve
at that one point, in the manner of the diagonal black line in the diagram. So
we have a definition: The slope of the curve at a point is the slope of the
tangent line at that point.
Now we can find the slope of the tangent line. We approximate it by drawing
a little right triangle (in gray in the diagram). One vertex is at (x,y) on the
curve. Another is on the curve, but a small distance delta x to the right of
(x,y). This point has not only a different x value (x+delta x), but a different
y value (y+delta y). The third vertex is the right angle, as in the diagram.
The hypotenuse of this right triangle has almost the same slope (delta y/delta
x) as our tangent line. We need to take a limit (see Limits). We make delta x get smaller and smaller, and
then delta y/delta x gets closer and closer to the slope of the tangent line.
Notice that we can never let delta x become zero, because then we would have
0/0 which is illegal (see Two Equals One?).
We call this limit the derivative (dy/dx) of the curve at (x,y). Here is
the definition of a derivative:
. And this is
the slope. Often, this is not difficult to calculate.
Now, why would you want to know a slope of a curve? Well, we might be
graphing (or studying) distances that change over time. The vertical axis is
the distance, and the horizontal axis is the time. In that case, the slope is
the velocity (speed). The slope (derivative) is the rate at which the
distance changes. That is very important in physics and engineering. If we
graph the velocity as it changes over time, then the derivative is the
acceleration. Again, that is sometimes very important. I can even graph the
amount of money I have in the bank. The slope is the interest rate.
Not only can we find the derivative of a
function at a point, but we can often find the derivative for all x (or for a
range of x values). Here we see the graph of the function y=x^2 (where x^2
means x squared). Merely by applying the definition of the derivative (above)
we find that the derivative of x^2 is 2x. See the equations on the left. This
means that the slope of the curve is 2x. If x is 1 then the slope is 2; x is 10
then the slope is 20.
We can determine derivatives of other functions in similar ways. Here are a
few:
- dx/dx=1
- dx^2/dx=2x
- d(mx^3+nx^2+ox+p)/dx=3mx^2+2nx+o
- dsinx/dx=cos x (where x is in radians)
- dcosx/dx=-sin x
- de^x/dx=e^x
Also, it is very useful to realize that a function reaches a relative
(local) minimum or maximum where the derivative (slope) is zero. For y=x^2,
that is when x=0, which we could have guessed.
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