If and increases from to then the change in is give by , see Fig. 4.1 . The differential is defined as
The derivative can also be interpreted as the slope of a curve, see Fig. 4.2 . If the slope at a given point has an angle , we find that is . In other words, the line is tangent to the curve at .
The differential of a sum is the sum of differentials,
L&T, F.9.26-27
There exists a simple rule to calculate the differential of a product,
E.g., if ,
L&T, F.9.28-30
In the same way we can find a relation for the differential of a quotient,
E.g., if ,
L&T, F.9.33-36,7.5-18
Often we take a function of a function. In such a case, where we put , and find
This rule is sometimes expressed in words as “the derivative of the function, times the derivative of its argument”, and you may know it as
Example 4.1:
Find for .
Solution:
Put so ,
Example 4.2:
Find for .
Solution:
Put so ,
Example 4.3:
Given that , find the velocity and the acceleration .
Solution:
Using the definitions of velocity as rate of change of position, we find that , and with acceleration as rate of change of velocity, we have .
Example 4.4:
For simple harmonic motion (SHM) . Find the velocity and acceleration.
Solution:
Use the change rule for differentiation, ,
L&T, 9.8-13
When we wish to calculate the differential of an inverse function, i.e, a function such that , we can use our knowledge of the derivative of to find that of .
Example 4.5:
Find the derivative of .
Solution:
We use and calculate first,
Now , but the slope of the inverse sine is always positive. Thus
L&T, 9.24-31
At a maximum or minimum the slope is 0 so that . To find which case it is, we look at , which can easily be done from a plot of the slope.
Example 4.6:
Find all maxima and minima of and determine their character.
Solution:
We find that . For a maximum or minimum the slope must be 0. This happens for , i.e., . For that value of , . So the point , is a (and the only) maximum.
L&T, F.9.21-22
Higher derivatives are obtained by differentiation 2 or more times, , .
Example 4.7:
, , , , etc.
Example 4.8:
The equation for simple harmonic motion (SHM) is . Prove that satisfies this equation.
Solution:
We must differentiate twice, start with first derivative, , and find that
QED.
N.B.: SHM not studied here, but in next semester. The constants , can only be obtained with extra input.