Math 121 - Calculus for Biology I Fall Semester, 2003 Rules of Differentiation
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1. Applications with Power Law
2. Notation for the Derivative
3. Power Rule
4. Height of a Ball
5. Other Basic Rules of Differentiation
6. Differentiation of Polynomials
7. Worked Examples
8. References

The previous section showed the definition of a derivative. However, it is clear that using the definition of the derivative is not an efficient way to find derivatives. In this section we develop some rules for differentiation. This section covers the basic power rule for differentiation, additive and scalar multiplication rules, and applications to polynomials.

Applications with Power Law

In the allometric section and the computer labs, we saw that many biological applications are reasonably well modeled by a power law relationship. For example, the data from Altman and Dittmer [1] for the pulse, P, as a function of the weight, w, are approximated by the relationship

The pulse is in beats/min, and the weight is in kilograms. Below is a graph of this relationship.

The graph shows an initial steep decrease in the pulse as weight increases, but can one quantify how fast the pulse rate changes as a function of weight? Clearly, for small animals the pulse rate changes more rapidly than for large animals. The derivative of this allometric or power law model provides more details on the rate of change in pulse rate as a function of weight.

In the computer lab, we saw that the number of species of herpatofauna, N, on Caribbean islands could be related to the area of the island, A, by an allometric model approximated by

A model of this sort is important for obtaining information about biodiversity. A graph of this model is seen below.

 

Can we use this model to determine the rate of change of numbers of species with respect to a given increase in area? Again the derivative is used to help quantify the rate of change of the dependent variable, N, with respect to the independent variable, A.

Notation for the Derivative

We begin with an introduction to some of the notation that we will use. There are several standard notations for the derivative. The two most common are the ones founded by Newton and Leibnitz. For the function f(x), the Newtonian notation for the derivative is written as follows:

The notation that Leibnitz used was

We will use these notations interchangeably, depending on what we are trying to show.

Power Rule

The power rule for differentiation is given by the formula

Examples:

Use the power rule to find the derivatives of the following functions:

1. f(x) = x5	2. f(x) = x-3
3. f(x) = x1/3	4. f(x) = 1/x4
5. f(x) = 1/x1/2	6. f(x) = 3

Solutions:

1. Since n = 5, it follows from the power rule that f '(x) = 5x⁴.

2. Since n = -3, it follows from the power rule that f '(x) = -3x^-4.

3. Since n = 1/3, it follows from the power rule that f '(x) = 1/3x^-2/3.

4. Since n = -4, it follows from the power rule that f '(x) = -4x^-5.

5. Since n = -1/2, it follows from the power rule that f '(x) = -1/2x^-3/2.

6. Since n = 0, the power rule does not apply, but we know that the derivative of a constant is f '(x) = 0.

Examples:

Consider our applications at the beginning of this section. For the model on pulse rate, P = 200w^-0.25, we use the power law of differentiation (and the fact that the scalar 200 multiplying the function is uneffected by differentiation) to obtain

The negative sign shows the decrease in the pulse rate with increasing weight. An animal at 16 kg by the allometric model would have a pulse of about 100 (since 200*16^-1/4 is 100). The derivative indicates that the pulse rate is decreasing by -50/32 beats/min/kg, so a 17 kg animal should have a pulse rate near 100 - 50/32 = 98 beats/min.

For the biodiversity model, N = 3A^1/3, we can differentiate using the power law to obtain

This shows the rate of change of numbers of species with respect to the island area is increasing as the derivative is positive, but the increase gets smaller with increasing island area, since the area has the power -2/3, which puts the area in the denominator of this expression for the derivative.

Height of a Ball

If a ball is thrown vertically in the air and air resistance is ignored (and we assume that the initial height of the ball is 0), then the height of a ball satisfies the formula,

where v₀ is the initial velocity of the ball thrown vertically and g is the acceleration due to gravity. We saw in our previous work that the derivative of this height function is given by the velocity function, which satisfies

This example takes advantage of three rules of differentiation. First, the additive property of derivatives allows consideration of each of the terms in the height function separately. Each of these terms has a scalar multiplier and a power of t. Thus, they use the power rule of differentiation along with a property for scalar multiplication. Below we list the rules for addition and scalar multiplication when taking a derivative.

Other Basic Rules of Differentiation

The operation of differentiation is said to be linear, which means that you can bring out multiplicative constants and the derivative of the sum of two functions is the sum of the derivatives.

Scalar Multiplication Rule:

Assume that k is a constant and f(x) is a differentiable function, then

Additive Rule:

Assume that f(x) and g(x) are differentiable functions, then

 Example: Differentiation of polynomials.

Consider the polynomial

From our rules above, it is easy to see that the derivative is

Example:

Clearly other additive powers are handled similarly.

From our rules above, it is easy to see that the derivative is

Worked Examples are available to help with the homework problems.

References:

[1] P. L. Altman and D. M. Dittmer, eds. (1964). Biology Data Book. Federation of American Societies for Experimental Biology. pp. 234-235.