(a) In your own words, describe the steps required to differentiate a product of two functions. The description should be more than one step. Do not just write down a formula.
(b) Take a product of two functions whose derivative you know and use the steps in part (a) to calculate its derivative.
(c) If the answer you obtained using your description doesn't match the known answer, edit your description and try again. Repeat with other products of functions until you are happy that your description is correct.
(a) If y = f(x)g(x) then writing
\[
\frac{dy}{dx} = \frac{df}{dx}g(x) + f(x)\frac{dg}{dx}
\]
suggests a pattern for products of more than two functions. By writing y = f(x)g(x)h(x) as
\[ y = f(x)(g(x)h(x)) \]
use the product rule to show
\[
\frac{dy}{dx} = \frac{df}{dx}g(x)h(x) + f(x)\frac{dg}{dx}h(x) + f(x)g(x)\frac{dh}{dx}
\]
(b) Given
\[ f(x) = x^n \sin x \]
Evaluate the first derivative for n = 1, 2, 3, 4 to find a pattern for any n.
(a) Given u(x) and v(x), both differentiable at x = 0, with
\[ \begin{align}
u(0) &= 5 \qquad v(0) = -1 \\
u'(0) &= -3 \quad v'(0) = 2 \\
\end{align} \]
find
\[ \frac{d}{dx} \ u(x)v(x) \]
at x = 0.
(b) Given
\[ f(x) = (x^2+1)(x + \frac{1}{x}) \]
Find f '(x) by first multiplying out the product and then by using the product rule.
Can you think of some guidelines to help you decide when to use which approach?