A fraction that has the same term in its numerator and denominator is equal to 1, so we write
= 2x(1) + 1 (1) = 2x + 1Every time we take a derivative of a term with x in it, we multiply by the term , but because this is 1,
we ignore it. Suppose however, that we wanted to find out how y changes with respect to t (for time).
Then we would have
If we wanted to find out how y changes with respect to r, we would have
and if we wanted to find out how y changes with respect to y, we would have
This is how we really do differentiation. Remember the following:
When you have an equation of x in terms of y, and you want to find the derivative with respect to y, simply
differentiate. But if the equation is of y in terms of x, find and take its reciprocal to find . Go back
to our original example.