**Unformatted text preview: **x ) of P ? ⇒ 13. Find a cubic polynomial whose graph has horizontal tangents at (-2 , 5) and (2 , 3). ⇒ 3.3 The Product Rule Consider the product of two simple functions, say f ( x ) = ( x 2 + 1)( x 3-3 x ). An obvious guess for the derivative of f is the product of the derivatives of the constituent functions: (2 x )(3 x 2-3) = 6 x 3-6 x . Is this correct? We can easily check, by rewriting f and doing the calculation in a way that is known to work. First, f ( x ) = x 5-3 x 3 + x 3-3 x = x 5-2 x 3-3 x , and then f ± ( x ) = 5 x 4-6 x 2-3. Not even close! What went “wrong”? Well, nothing really, except the guess was wrong. So the derivative of f ( x ) g ( x ) is NOT as simple as f ± ( x ) g ± ( x ). Surely there is some rule for such a situation? There is, and it is instructive to “discover” it by trying to do...

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- Fall '07
- JonathanRogawski
- Math, Calculus, Derivative, 1–6