3.2: Product and Quotient Rules
We know how to take derivatives of sums, constants, and $x^n$. What about products?
Product Rule
\[\dfrac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)\]
Find the derivatives of the following:
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$f(x) = (2x^2 - 1)(x^3 + 3)$
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$f(x) = x^3(\sqrt{x} + 1)$
How about quotients?
Quotient Rule
\[\dfrac{d}{dx}\left[\dfrac{f(x)}{g(x)}\right] = \dfrac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}\]
You can remember this by thinking "low Dhigh minus high Dlow, square the bottom, away you go."
Differentiate (find the derivative) of the following:
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$f(x) = \dfrac{x}{2x-4}$
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$f(x) = \dfrac{x^2 + 1}{x^2-1}$
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$f(x) = \dfrac{\sqrt{x}}{x^2+1}$
The annual sales (in millions of dollars per year) of a DVD recording of a hit movie $t$ years from the date of release is given by \[S(t) = \dfrac{5t}{t^2 + 1}\]
Find a function for the rate at which the annual sales are changing at time $t$.