In 1986, Unit 4 of Chernobyl nuclear power plant exploded. The accident changed the lives of millions of people.
When the explosion happened, big amounts of radioactive elements were released into the air and spreading far outside of the vicinity of Chernobyl. These substances decay exponentially, but at a very slow rate. For example, after 24.000 years there will still remain half of the quantity of plutonium-239.
As several of the substances are harmful, it's interesting to know how quickly they decay. Scientists have been able to determine the exponential curves describing this process.
Knowing the rate of change, the derivative, of such a curve lets us predict how much will still be left at some point later in time.
Let's say we have a population of some nasty bacteria, which double every hour.
The function which describes "doubling every hour" is . As the derivative is just the rate of change, one might feel inclined to say that the derivative of an exponential is itself: when we have four bacteria, the next hour they have increased by four, and so on.
However, the derivative is not limiting itself to change per hour. Estimating the instantaneous rate of change by making the time intervals infinitely small, we get a slightly different value. Actually, the derivative of is itself, times a constant:
This goes for all exponential functions. The mathematicians' favorite is , for which this constant is , making the function and its derivative identical.
To calculate the derivative of the exponential function we revisit standard limits (they are very handy!). Using the definition of the derivative, we obtain:
As has no in it, we can move it outside of the limit. The limit remaining might be familiar:
So, the derivative of is itself! We use this for calculating the derivatives of all other exponential functions.
The inverse of is , whose derivative can be found by letting . It turns out to be . Don't worry if you have no idea why; the neat little proof awaits in the lecture notes.
Derivatives of exponential functions
The exponential function has a very handy property: it is its own derivative. We will see how we can use this to differentiate any exponential function.
As we saw in the introduction, all exponential functions have exponential derivatives. The derivative and the function also have the same base. The derivative of any differential function thus looks like this:
The difference between the exponential functions' derivatives is the constant that appears. For , .
Functions of the type pop up everywhere, as they are so handy to deal with. That is its own derivative also implies that it's its own antiderivative. The practicality of this can hardly be overestimated.
Deriving the derivative of
Soon we will have a look at how we can transform to and subsequently differentiate it. First, we derive the derivative of . This can be done by using a nifty standard limit and the definition of the derivative.
If you feel shaky on properties of exponential functions, now is a good time to quickly review them.
So let's get to it. The derivative of is derived as follows:
Notice that has no in it. Thus, we can move it to outside of the limit. You might recognize the remaining standard limit:
The derivative of
So, what is that constant appearing in front as we differentiate ?
Finally, the time has come to reveal its true identity. The comes from the fact that we can write:
Now, notice that in the exponent, we have an inner function , where is a constant.
Thus, using the chain rule, the inner derivative popping out is just , and we get:
There it is: , and:
Derivatives of logarithmic functions
When studying the power rule, we saw that:
Applying our knowledge about antiderivatives, we see that for , we have:
This works perfectly fine for all powers except one, namely , which would yield a denominator with value 0.
On the one hand, it makes sense that this case breaks the general form, recalling that the derivative of any constant is zero, and that .
On the other hand, it raises an intriguing question. What will be an antiderivative of ?
The answer may come as a surprise, but using implicit differentiation, we can prove that it is , the natural logarithm of , that we need to differentiate in order to get .
The derivative of logarithms with any base
The derivative of logarithmic functions
Let . Then , and so:
Rearranging the expression and using gives:
The derivative of the natural logarithm
We have proven the derivative of the logarithmic function of any base . Now let's examine what happens when we have Euler's number as the base ().
Consequently, we have solved the mystery of finding the function whose derivative is :