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Home Differentiation Derivatives of Rational Functions Theorem 4: Product Rule
 
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Theorem 4: Product Rule

THEOREM 4 (Product Rule)

Suppose u and v depend on x. Then for any value of x where du/dx and dv/dx exist,

02_differentiation-104.gif

PROOF

Let y = uv, and let Δx ≠ 0 be infinitesimal.

02_differentiation-105.gif

Δu is infinitesimal by the Increment Theorem, whence

02_differentiation-106.gif

So

02_differentiation-107.gif

The Constant Rule is really the special case of the Product Rule where v is a constant function of x, v = c. To check this we let v be the constant c and see what the Product Rule gives us:

02_differentiation-108.gif

This is the Constant Rule.

The Product Rule can also be used to find the derivative of a power of u.
Home Differentiation Derivatives of Rational Functions Theorem 4: Product Rule

Last Update: 2010-11-25