Derivatives. in physics. Why Do We Need Derivatives?. In physics things are constantly changing.
y(x) = 3
It doesn’t change. It is a constant value of 3 everywhere. .. but how would we describe your response mathematically?
The slope of the line equals Δy/Δx
y(x) = 3x + 1
Using our definition for the change in y(x) with respect to x from the previous slide, we get:
Δy/Δx = 3
The function y changes by 3 units between x=2 and x=3.
…it is just the slope of the line!!!
Now let us consider a more complicated function other than a straight line. Consider the graph to the right. How would you describe how the function y(x) changes at any given value of x?
Well, from what you just learned, the change in the function y(x) with respect to x should be given by the slope of a straight line. But what is the straight line???
Slope of the tangent line at point X gives the change in y(x) with respect to x.:
As you will see……the change in y(x) with respect to x at point X is given by the slope of the tangent line at point X!
The word “tangent line” describes a line that intersects or touches a curve at only one point. At any given point on a smooth curve, there is only one unique tangent line and therefore there is only one value for the slope of that tangent line at that point.
To determine the slope of the tangent line, let us draw a line that intersects the curve at both point X and point X + h.
What is the slope of this line? From before, the slope of the line will be Δy/Δx:
The pink line is (obviously) not tangent, so let us draw another line that intersects the curve at both point X and point X + h.
This maroon line is still not the tangent line at point X, but we can make it look more like the tangent line if we make the h a smaller value:
If you calculate the slope for this second line, it will have the same form as the above equation, except now h is a smaller value and therefore y(X+h) will be a different value.
h goes to zero!
h gets even smaller
h gets smaller
Let animation run!!!
Slope of tangent line =
is what we call the derivative of y with respect to x.
We use the symbol
to represent the slope of the tangent line.
Therefore, the term “derivative” just represents how the function y(x)instantaneously changes with respect to the variable x. As h goes to zero, Δy/Δx becomes dy/dx.
You could see that as h approached zero, the slope of the line became closer to that of the slope of the tangent line at x.
Tangent line at x = 5 has a slope of 10. Therefore, the function y(x) has an instantaneous slope of 10 units at x=5.
y = x2
Let us use this result to determine the derivative at x = 5. Since the derivative of y(x)=x2 equals 2x, then the derivative at x = 5 is 2*5 = 10. Therefore, the slope of the tangent line that passes through x = 5 has a slope of 10!
y = x2
The slope of the tangent line at x = -4 is -8.
The slope of the tangent line at x = -9 is -18.
The slope of the tangent line at x = 0 is 0.
dy/dx = 2x
Definition of derivative
Substituted in the expression for y(x)
Terms that survived after some terms canceled
Divided each term by h
These terms survived after h went to zero
“Repetition is good for the soul,” therefore, let us now do another example. Consider the formula y =4x3-15x2+20. This function is graphed to the right. 1. Find the derivative (use the definition…YES…ALL of you)2. Plot a graph of the derivative.Go…
dy/dx = 12x2-30x
The slope of the tangent lines should be zero at these points.
Between x = 0 and x = 2.5, y(x) should have a negative slope.
Let us compare the two graphs and some of their features:
In this region, dy/dx should be positive.
Indeed, between x=0 and x=2.5, dy/dx is negative in value.
Indeed, dy/dx has positive values between x=-2 and x=0.
Indeed, at x = 0 and x = 2.5, dy/dx equals zero.
When it comes to functions of the form: y(x) = Axn, where “A” is just a numerical constant and “n” is an integer, positive or negative. Plugging this expression into the definition for the derivative, you will find that:
dy/dx = nAxn-1
Using this shortcut to calculate the derivative of y(x) = x2, we get:
dy/dx = 2*x2-1 = 2x. This is exactly what we got earlier in example 1.
In our second example, we took the derivative of y =4x3-15x2+20. Using the shortcut, we find: dy/dx = (3)4x2 – (2)15x+ 0 = 12x2 – 30
That’s the same thing that we had before….HOORAY! The shortcut works!
What are these symbols?
“f” and “g” are functions of x: f(x) and g(x).
“c” represents a constant numerical value and therefore is not a function of x.
“n” represents an integer number, positive or negative.
f’ is shorthand for df/dx. Likewise, g’ is shorthand for dg/dx.
Mathematically, we can represent this as:
This is just ONE EXAMPLE of a change in physics…throughout this course we will encounter MANY changes…thus MANY places to use the derivative