The most difficult part of finding a derivative is evaluating the limit involved in the definition of the derivative at a point. Often there is some algebra and simplifying involved as the example below demonstrates.
Problem Suppose the function g(x) is given by
Use the definition of the derivative at a point to compute g´(3).
Solution The definition of the derivative of g(x) at x = 3 is
The function value g(3) is calculated to be
The function value g(3 + h) must be calculated carefully.
Form the difference quotient and simplify:
The derivative is completed by taking the limit ash approaches zero,
The derivative of g(x) evaluated at x = 3,g´(3), is 11.
A tangent line to a function is a line that looks most like the function at a point. In common terms, it just grazes the function.
To find its equation, we need to locate the point where the two meet as well as the slope of the function at that point. Then we can use the slope-intercept form or point-slope form of a line to get the equation.
Find the equation of the tangent line to
at x = 3.
Since this problem is asking for the equation of a line, let’s start with the point-slope form
This requires a point (x1, y1) and slope m. We’ll use the function to get the point and the derivative to get the slope of the tangent line.
Find the point: We are given a point x = 3. To find the corresponding y value, put the x value into the function
Find the slope of the tangent line: We need h′(3) to get the slope of the tangent line. We’ll use the Power Rule to take the derivative,
The slope of the tangent is
Write the equation of the tangent line: Putting the point (3, 10) and the slope 9 into the line yields
If you are asked to write this in slope-intercept form, you’ll need to solve this for y to give
If you graph h(x) and the tangent line together, it should be obvious that your tangent line is correct (ie. tangent).
The instantaneous rate of change is calculated to find how fast one quantity changes with respect to another.
The instantaneous rate of change of f (x)with respect to x at x = a is
To apply this definition, you need to identify the point a at which the rate is to be calculated. Then the function values f (a) and f (a+h) are calculated and simplified. Finally, these are substituted into the limit so that it evaluated.
Example 1 Find the instantaneous rate of change of at .
Solution Start by calculating the two function values.
Once you have the function values, substitute them into the definition for instantaneous rate of change.
Example 2 Find the instantaneous rate of change of at .
Solution The function values are
Now put these into the limit definition of instantaneous rate of change.
Many students struggle with slopes of tangent lines versus slopes of secant lines. In the example below, I find these slopes and use them to compute the equation of a tangent line and the equation of a secant line.
The secant line is the red line to the right that passes through two points on the curve. The tangent line is the green line that just grazes the curve at a point.
We are now in the real meat of calculus….rates and derivatives. Essentially, rates are simply slopes. Depending on the field you are working in or the representation of the data or function, it might have a different terminology. Keeping all of these straight is the biggest challenge in the class. Let’s try to put what we know into a table.
Average Rate of Change of f(x) from x=a to x=b
Slope between (a,f(a)) and (b,f(b))
Slope of the Secant Line from x=a to x=b
Instantaneous Rate of Change of f(x) at x=a
Slope between (a,f(a)) and (a+h,f(a+h)) where h is very small
Slope of the Tangent Line on f(x) at x=a
If it feels like we use slope to calculate everything…you are right. The only difference is where the two points are located. If they are located fairly far apart, then we are calculating an average rate of change. If they are located an infinitesimal amount apart, then we are calculating the instantaneous rate of change.
But here is where it gets hazy…if the representation of the function does not allow us to pick the points infinitely close together, we may approximate the instantaneous rate of change with an average rate of change between two points that are as close as the representation allows. I think this point is bothering many of you. It occurs most often when we try to find the instantaneous rate from a data table. In that case, the points are where they are at and we can pick pairs that are closer and closer together. We pick them as close as we can near the place we want the instantaneous rate of change and say we are estimating the instantaneous rate of change.
If we are given a graph, we can draw a secant line between the points to calculate the average rate of change. The only estimating done in this case is estimating where the points are on the graph. Our eyes are only so good in reading off the values. For the instantaneous rate of change, we draw the tangent line where we want the rate and eyeball its slope. Again, since we are making educated guesses about the slopes, the numbers are estimates based on our ability to read the slope.
If we are given the function’s formula, we can calculate the average rate or the instantaneous rate exactly. For the average rate of change of f(x) over x = a to x = b, we calculate .
For the instantaneous rate of change, we use the function’s formula to calculate the limit .
The estimating comes from the fact that we may not be able to find two points infinitesimally close or the fact that we cannot calculate the slope perfectly from how the function is given to us.
Another cautionary note…different disciplines call the derivative by different names. Mathematicians call it the derivative of course, but in physics it might be called the instantaneous rate. In finance and economics, it will be called the marginal function. And in other fields you’ll here other terms.