Product Rule & Quotient Rule
The Product Rule
In calculus, the product rule is a formula used to calculate the derivative of the product of two or more functions, like for example f(x) = x*sin(x) or f(x) = 4x*cos(x). This is the formula for the product rule:
Product Rule Formula
d (uv) = vdu + udv
dx dx dx
The product rule formula is used when differentiating a product of two functions. What this formula says is that the derivative of the function u * v is equal to the result of v times the derivative of u plus the result of u times the derivative of v.
Example: The Product Rule
Find the derivative of f(x) = x2cos(x).
In this problem we have the product of two functions: x2 and cos(x). To find the derivate d/dx, we'll use the product formula.
We'll let u = cos(x) and v = x2.
According to the product rule, we need to also find the derivative of u and the derivative of v.
Example: The Chain Rule
Find the derivative of f(x) = cos(x2).
We'll let f(x) = cos(x) and g(x) = x2
By the chain rule we have that the derivative of f(g(x)) is f'(g(x))g'(x)
f'(x) = -sin(x)
g'(x) = 2x
Therefore, we have that the derivative of f(x) = cos(x2) equals:
[cos(x2)]' = -2xsin(x2)
The Quotient Rule
d/dx(u/v) = v(du/dx) - u(dv/dx)
v2
Example: The Quotient Rule
If y = x3 , find dy/dx
x + 4
Let u = x3 and v = (x + 4). Using the quotient rule, dy/dx =
(x + 4)(3x2) - x3(1) = 2x3 + 12x2
(x + 4)2 (x + 4)2