Given two functions f and g, there are a few easy ways to build other functions by ordinary arithmetic operations such as sum, multiplication, and division.
- We define the sum of functions as follows:
By the same token, we define the difference of two functions. To be precise, this makes sense on the intersection of the two domains, but we will not be bothered by such details. - Given a number α and a function f, we can apply multiplication by a constant:
- Given two functions f and g, we define the product of functions:
- Finally, we may use division of functions:
Again, this makes sense on a common domain, where both functions f and g are defined, provided that g(x) ≠ 0.
If we are able to find the derivative of f and g, the following theorem shows how to find the derivative of functions defined by the mechanisms above.
THEOREM 2.7 Let f and g be functions and α be a real number. If f and g are defined and differentiable at x0, then
The last result also requires g(x0) ≠ 0.
The following examples illustrate the application of the theorem.
Example 2.20 A polynomial function is basically a sum of monomials, obtained by multiplying a number and an integer power of x. Then, to find the derivative of a polynomial we can use the first two results as follows. Consider
The derivative of the first term is
The same approach yields the derivatives of the second and third terms. The last term is just a constant, and its derivative is zero. Putting everything together, we have
Example 2.21 Let us illustrate the product of functions. Consider the product of a polynomial and an exponential:
We may easily take the derivative of each factor of the product:
Applying the result for the product of functions, we obtain
Example 2.22 Finally, let us illustrate the case of a rational function, obtained by dividing two polynomials:
We should break down the overall task into smaller pieces. First we compute the derivatives of numerator and denominator:
Then we put everything together:
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