Constants in Differentiation | Calculus Made Easy by Silvanus Thompson

In our equations we have regarded as growing, and as a result of being made to grow also changed its value and grew. We usually think of as a quantity that we can vary; and, regarding the variation of as a sort of cause, we consider the resulting variation of as an effect. In other words, we regard the value of as depending on that of . Both and are variables, but is the one that we operate upon, and is the “dependent variable.” In all the preceding chapter we have been trying to find out rules for the proportion which the dependent variation in bears to the variation independently made in .

Our next step is to find out what effect on the process of differentiating is caused by the presence of constants, that is, of numbers which don’t change when or change their values.

Added Constants

Let us begin with some simple case of an added constant, thus:

Example 5.1. Let Just as before, let us suppose to grow to and to grow to .

Then: Neglecting the small quantities of higher orders, this becomes Subtract the original , and we have left:

So the has quite disappeared. It added nothing to the growth of , and does not enter into the derivative. If we had put , or , or any other number, instead of , it would have disappeared. So if we take the letter , or , or to represent any constant, it will simply disappear when we differentiate.

If the additional constant had been of negative value, such as or , it would equally have disappeared.

In general:

where and are constants.

Multiplied Constants

Take as a simple experiment this case:

Example 5.2. Let .
Then on proceeding as before we get: Then, subtracting the original , and neglecting the last term, we have

Let us illustrate this example by working out the graphs of the equations and , by assigning to a set of successive values, , , , , etc., and finding the corresponding values of and of . These values we tabulate as follows:

	0	1	2	3	4	5
	0	7	28	63	112	175	7	28	63
	0	14	28	42	56	70

Now plot these values to some convenient scale, and we obtain the two curves, Figs. 5.1 and 5.2.

Carefully compare the two figures, and verify by inspection that the height of each point on the graph of the derivative (Fig. 5.2) is proportional to the slope of the graph¹ of the original function (Fig. 5.1) at the corresponding value of . To the left of the origin, where the graph of the original function slopes negatively (that is, downward from left to right), the corresponding vertical coordinates of points on the graph of the derivative are negative.

Now if we look back at Example 4.1, we shall see that simply differentiating gives us . So that the derivative of is just times as big as that of . If we had taken , the derivative would have come out eight times as great as that of . If we put , we shall get

If we had begun with , we should have had . So that any mere multiplication by a constant reappears as a mere multiplication when the thing is differentiated. And, what is true about multiplication is equally true about division: for if, in the example above, we had taken as the constant instead of , we should have had the same come out in the result after differentiation.

where and are constants.

Combining these two rules:

where , , and are constants.

Some Further Examples

The following further examples, fully worked out, will enable you to master completely the process of differentiation as applied to ordinary algebraical expressions, and enable you to work out by yourself the examples given at the end of this chapter.

Example 5.3. Differentiate .

Solution. is an added constant and vanishes (see here).

We may then write at once or

Example 5.4. Differentiate .

Solution. The term vanishes, being an added constant; and as , in the index form, is written , we have or

Example 5.5. If find the derivative of with respect to .

Solution. As a rule an expression of this kind will need a little more knowledge than we have acquired so far; it is, however, always worth while to try whether the expression can be put in a simpler form.

First, we must try to bring it into the form some expression involving only.

The expression may be written

Squaring, we get which simplifies to or that is

hence²

Example 5.6. (a) The volume of a cylinder of radius and height is given by the formula . Find the rate of variation of volume with the radius when in and in (b) If , find the dimensions of the cylinder so that a change of in in radius causes a change of in in the volume.

Solution. (a) The rate of variation of with regard to is

If in and in, this becomes . It means that a change of radius of inch will cause a change of volume of in. This can be easily verified, for the volumes with and are in and in respectively, and .

(b) If we must have

Example 5.7. The reading of a Féry’s Radiation pyrometer is related to the Centigrade temperature of the observed body by the relation where is the reading corresponding to a known temperature of the observed body.

Compare the sensitiveness of the pyrometer at temperatures C, C, C, given that it read when the temperature was C.

Solution. The sensitiveness is the rate of variation of the reading with the temperature, that is . The formula may be written and we have

When C, C and C, we get , and , respectively. The sensitiveness is approximately doubled from C to C, and becomes three-quarters as great again up to C.

Exercises

Differentiate the following:

Exercise 5.1. .

Solution

The constant is added and disappears during the process of differentiation.

Exercise 5.2. .

Solution

is an added constant and vanishes during the process of differentiation.

Exercise 5.3. .

Solution

is an added constant and vanishes.

Exercise 5.4. .

Solution

Exercise 5.5. .

Solution

Exercise 5.6.

Solution

Make up some other examples for yourself, and try your hand at differentiating them.

Exercise 5.7. If and be the lengths of a rod of iron at the temperatures C. and C. respectively, then . Find the change of length of the rod per degree centigrade.

Solution

Exercise 5.8. It has been found that if be the candle power of an incandescent electric lamp, and be the voltage, , where and are constants.

Find the rate of change of the candle power with the voltage, and calculate the change of candle power per volt at , and volts in the case of a lamp for which and .

Solution

So for and ,

When volts

Exercise 5.9. The frequency of vibration of a string of diameter , length and specific gravity , stretched with a force , is given by

Find the rate of change of the frequency when , , and are varied singly.

Solution

When is varied, we write

then

When is varied

Exercise 5.10. The greatest external pressure which a tube can support without collapsing is given by where and are constants, is the thickness of the tube and is its diameter. (This formula assumes that is small compared to .)

Compare the rate at which varies for a small change of thickness and for a small change of diameter taking place separately.

Solution

The problem is asking us to calculate and and find their ratio

Exercise 5.11. Find, from first principles, the rate at which the following vary with respect to a change in radius:

the circumference of a circle of radius ;

the area of a circle of radius ;

the lateral area of a cone of slant dimension ;

the volume of a cone of radius and height ;

the area of a sphere of radius ;

the volume of a sphere of radius .

Answer

, , , , , .

Solution

(a) The circumference of a circle is given by

Therefore

(b) The area of a circle is given by

Therefore

Therefore,

(d) The volume of a cone of radius and height is

Therefore

(e) The area of a sphere of radius is

(f) The volume of a sphere of radius is

Therefore

Exercise 5.12. The length of an iron rod at the temperature being given by , where is the length at the temperature , find the rate of variation of the diameter of an iron tire suitable for being shrunk on a wheel, when the temperature varies.

Answer

, , , , , .

Solution

Since , we have

You have now learned how to differentiate powers of . How easy it is!

The Added Constants and Multiplied Constants: Their Effects on Derivatives

Added Constants

Multiplied Constants

Some Further Examples

Exercises

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