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Properties of the derivative

10 November, 2015 - 14:23
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It follows immediately from the definition of the derivative that multiplying a function by a constant multiplies its derivative by the same constant, so for example since we know that the derivative of t^2/2 is t, we can immediately tell that the derivative of t^2 is 2t, and the derivative of t^2/17 is 2t/17.

Also, if we add two functions, their derivatives add. To give a good example of this, we need to have another function that we can differentiate, one that isn't just some multiple of t^2. An easy one is t: the derivative of t is 1, since the graph of x=t is a line with a slope of 1, and the tangent line lies right on top of the original line.

Example

The derivative of 5t^2+2t is the derivative of 5t^2 plus the derivative of 2t, since derivatives add. The derivative of 5t^2 is 5 times the derivative of t^2, and the derivative of 2t is 2 times the derivative of t, so putting everything together, we find that the derivative of 5t^2+2t is (5)(2t)+(2)(1)=10t+2. The derivative of a constant is zero, since a constant function's graph is a horizontal line, with a slope of zero. We now know enough to differentiate any second order polynomial.

Example

An insect pest from the United States is inadvertently released in a village in rural China. The pests spread outward at a rate of s kilometers per year, forming a widening circle of contagion. Find the number of square kilometers per year that become newly infested. Check that the units of the result make sense. Interpret the result.

Let t be the time, in years, since the pest was introduced. The radius of the circle is r=st, and its area is a=\pi r^2=\pi(st)^2. To make this look like a polynomial, we have to rewrite it as a=(\pi s^2)t^2. The derivative is

\begin{align*} \dot{a} &=(\pi s^2)(2t) \\ \dot{a}&=(2\pi s^2)t \end{align*}

The units of s are km/year, so squaring it gives \textrm{km}^2/\textrm{year}^2. The 2 and the \pi are unitless, and multiplying by t gives units of \textrm{km}^2/\textrm{year}, which is what we expect for \dot{a}, since it represents the number of square kilometers per year that become infested.

Interpreting the result, we notice a couple of things. First, the rate of infestation isn’t constant; it’s proportional to t, so people might not pay so much attention at first, but later on the effort required to combat the problem will grow more and more quickly. Second, we notice that the result is proportional to s^2. This suggests that anything that could be done to reduce s would be very helpful. For instance, a measure that cut s in half would reduce \dot{a} by a factor of four.

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