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How to Find Derivatives using the Product Rule

The product rule is the rule you use when you have a product of two or more functions.

Let u(x) and v(x) be two functions of x. The product rule tells you how to differentiate the product of these two functions. The product is the new function u(x) v(x). To simplify it, you just write u and v for the functions u(x) and v(x), but it’s important to remember that these are functions of x.

Formula

The Product Rule

(uv) = uv + uv,

where u = u(x) and v = v(x).

Example 1

Differentiate the expression x22x

Looking at the expression, you decide that u = x2 and v = 2x. That gives you u = 2x and v = 1 2x, and the differentiation is (x22x) = (x2) 2x + x2 (2x) = 2x 2x + x2 1 2x = 2x2x + x2 2x = 2x2x 2x 2x + x2 2x = 4x2 2x + x2 2x = 5x2 2x

Example 2

Differentiate the expression ex (2x3 + 3x)

Here, you decide to say that u = ex and v = 2x3 + 3x. That gives you u = ex and v = 6x2 + 3, and the differentiation becomes

[ex (2x3 + 3x)] = (ex) (2x3 + 3x) + ex (2x3 + 3x) = ex (2x3 + 3x) + ex (6x2 + 3) = 2x3ex + 3xex + 6x2ex + 3ex = ex (2x3 + 6x2 + 3x + 3)

[ex (2x3 + 3x)] = (ex) (2x3 + 3x) + ex (2x3 + 3x) = ex (2x3 + 3x) + ex (6x2 + 3) = 2x3ex + 3xex + 6x2ex + 3ex = ex (2x3 + 6x2 + 3x + 3)

Example 3

Differentiate the expression (x2 + x 1) ln x

Here, you decide that u = x2 + x 1 and v = ln x. That gives you u = 2x + 1 and v = 1 x, and the differentiation becomes

[ (x2 + x 1) ln x] = (x2 + x 1) ln x = + (x2 + x 1) (ln x) = (2x + 1) ln x = + (x2 + x 1) 1 x = 2x ln x + ln x + x + 1 1 x

[ (x2 + x 1) ln x] = (x2 + x 1) ln x + (x2 + x 1) (ln x) = (2x + 1) ln x + (x2 + x 1) 1 x = 2x ln x + ln x + x + 1 1 x

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