• We have learned how to differentiate a polynomial function expressed in its fully expanded and simplified form. For example, \(f(x)=ax^2+bx+c\).

In This Module

• We will learn how to differentiate a polynomial in factored form. For example, \(f(x) = (x^2 + 2)(x + 5)\).
• We will determine a method of finding the derivative of \(f(x)=p(x)q(x)\), which does not require us to expand and simplify before differentiating.

Let's use first principles to find a general expression for the derivative of the product of two differentiable functions, \(p(x)\) and \(q(x)\).

Let \( f(x) = p(x)q(x) \). Then, by the definition of the derivative,

Now, consider the expression \(- p(x)q(x+h) + p(x)q(x+h)\).

This expression equals zero, so we can insert this expression into the limit calculation without changing its value.

\(f'(x) = \displaystyle \lim_{h \rightarrow 0}\left[q(x+h)\left(\dfrac{p(x+h) - p(x)}{h}\right) + p(x)\left(\dfrac{q(x+h) - q(x)}{h}\right)\right]\)

Assuming that each limit in the following sums exist, we can apply the properties of limits to simplify the following,

Note:

When differentiating expressions, we must always consider the domain on which the expression is differentiable. If \(p(x)\) or \(q(x)\) are not differentiable at the point \(a\) , then \(f(x)\) may not be differentiable at that point.

For example, suppose \(f(x) = x^2\lvert x\rvert\). Here, \(q(x)=\lvert x\rvert\) and \(q(x)\) has a corner at \(x=0\), so \(q(x)\) is not differentiable at \(x=0\). Therefore, \(f(x)\) is also not differentiable at \(x=0\).