Problem: Find the derivative of
Solution: The function is given by a formula, so we use rules to differentiate. There is one slightly unusual thing here, in the formula we see two parameters p and q whose values are unknown. However, this is not really a problem. When parameters are present, it is understood that they represent two fixed numbers, and although we do not know exactly what numbers are these, we do know that they do not change. Thus in our process of differentiation they play the same role as any other constant present in the formula (for instance the number 13).
In other words, we just imagine some concrete numbers instead of p and q and thing of what we would do in such a situation, then we do it with letters instead of numbers. Let's do it.
The last operation performed is the logarithm, so we face a
composition with logarithm as the outer function, so it has to be differentiated
first (using the chain rule). We recall that derivative of
Now the last operation is multiplication, so we apply the product rule.
In the first term we differentiate ordinary power, the second derivative is again a composition, this time a very simple one and the chain rule tells us what to do, we differentiate the hyperbolic cosine first.
What is the domain of this derivative? We start with the domain of the given
function. Logarithm requires its argument to be positive. What do we know of
the expression inside? Cosine hyperbolic accepts anything, so no restriction
there, and its values are always at least 1. The power is more selective.
Since we do not know what value the parameter p takes, we have
account for the worst possible case, when it is just some real number, and
require that
