The Product Rule for Differentiation

If you had to differentiate f(x) = (3x + 2)(x – 1), how would you start?

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Examine the original function f(x) = (3x + 2)(x – 1)

It is a product of two functions. What are they?

g(x) = and h(x) =

Since f(x) = g(x) h(x) does f /(x) g /(x) h /(x) ?

)(' xg )(' xh

)3)(1()1)(23()(' xxxf

16)(' xxf

We can derive this function another way as shown below. Where do the parts come from?

When we expanded, we determined the derivative to be

To differentiate a function f(x) which is the product of two functions g(x) and h(x) you……

Multiply the first function by the derivative of the second function

then add

The product of the second function and the derivative of the first

If f(x) = g(x)h(x), then

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f(x) = g(x)h(x) f /(x) = g(x) h /(x) + h(x) g /(x)

Example 1 Differentiate f(x) = x2 sin x

Let g(x) = and h(x) =

g /(x) = h /(x) =

f /(x) = In its simplest form:

f /(x) =

Example Two - Differentiate

132 2 xxxy