The Product Rule
The Product Rule
We’re going to find the derivative of products of functions, forexample:
The Product Rule
We’re going to find the derivative of products of functions, forexample:
f (x) = x sin x
The Product Rule
We’re going to find the derivative of products of functions, forexample:
f (x) = x sin x
Let’s consider a function:
The Product Rule
We’re going to find the derivative of products of functions, forexample:
f (x) = x sin x
Let’s consider a function:
f (x) = u(x)v(x)
The Product Rule
We’re going to find the derivative of products of functions, forexample:
f (x) = x sin x
Let’s consider a function:
f (x) = u(x)v(x)
We apply the definition:
The Product Rule
We’re going to find the derivative of products of functions, forexample:
f (x) = x sin x
Let’s consider a function:
f (x) = u(x)v(x)
We apply the definition:
f ′(x) = lim∆x→0
The Product Rule
We’re going to find the derivative of products of functions, forexample:
f (x) = x sin x
Let’s consider a function:
f (x) = u(x)v(x)
We apply the definition:
f ′(x) = lim∆x→0
f (x + ∆x) − f (x)
∆x
The Product Rule
We’re going to find the derivative of products of functions, forexample:
f (x) = x sin x
Let’s consider a function:
f (x) = u(x)v(x)
We apply the definition:
f ′(x) = lim∆x→0
f (x + ∆x) − f (x)
∆x
= lim∆x→0
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
The Product Rule
We’re going to find the derivative of products of functions, forexample:
f (x) = x sin x
Let’s consider a function:
f (x) = u(x)v(x)
We apply the definition:
f ′(x) = lim∆x→0
f (x + ∆x) − f (x)
∆x
= lim∆x→0
������
�����: f (x+∆x)
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
The Product Rule
We’re going to find the derivative of products of functions, forexample:
f (x) = x sin x
Let’s consider a function:
f (x) = u(x)v(x)
We apply the definition:
f ′(x) = lim∆x→0
f (x + ∆x) − f (x)
∆x
= lim∆x→0
u(x + ∆x)v(x + ∆x) −�����: f (x)
u(x)v(x)
∆x
The Product Rule
We’re going to find the derivative of products of functions, forexample:
f (x) = x sin x
Let’s consider a function:
f (x) = u(x)v(x)
We apply the definition:
f ′(x) = lim∆x→0
f (x + ∆x) − f (x)
∆x
= lim∆x→0
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
The Product Rule
The Product Rule
f ′(x) = lim∆x→0
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
The Product Rule
f ′(x) = lim∆x→0
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
Now we apply an algebraic trick: add and subtract u(x)v(x + ∆x).
The Product Rule
f ′(x) = lim∆x→0
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
Now we apply an algebraic trick: add and subtract u(x)v(x + ∆x).
u(x + ∆x)v(x + ∆x) − u(x)v(x + ∆x) + u(x)v(x + ∆x) − u(x)v(x)
∆x
The Product Rule
f ′(x) = lim∆x→0
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
Now we apply an algebraic trick: add and subtract u(x)v(x + ∆x).
u(x + ∆x)v(x + ∆x) − u(x)v(x + ∆x) + u(x)v(x + ∆x) − u(x)v(x)
∆x
The Product Rule
f ′(x) = lim∆x→0
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
Now we apply an algebraic trick: add and subtract u(x)v(x + ∆x).
u(x + ∆x)v(x + ∆x) − u(x)v(x + ∆x) + u(x)v(x + ∆x) − u(x)v(x)
∆x
The Product Rule
f ′(x) = lim∆x→0
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
Now we apply an algebraic trick: add and subtract u(x)v(x + ∆x).
u(x + ∆x)v(x + ∆x) − u(x)v(x + ∆x) + u(x)v(x + ∆x) − u(x)v(x)
∆x
The Product Rule
f ′(x) = lim∆x→0
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
Now we apply an algebraic trick: add and subtract u(x)v(x + ∆x).
u(x + ∆x)v(x + ∆x) − u(x)v(x + ∆x) + u(x)v(x + ∆x) − u(x)v(x)
∆x
= lim∆x→0
v(x + ∆x) [u(x + ∆x) − u(x)] + u(x) [v(x + ∆x) − v(x)]
∆x
The Product Rule
f ′(x) = lim∆x→0
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
Now we apply an algebraic trick: add and subtract u(x)v(x + ∆x).
u(x + ∆x)v(x + ∆x) − u(x)v(x + ∆x) + u(x)v(x + ∆x) − u(x)v(x)
∆x
= lim∆x→0
v(x + ∆x) [u(x + ∆x) − u(x)] + u(x) [v(x + ∆x) − v(x)]
∆x
= lim∆x→0
v(x + ∆x)u(x + ∆x) − u(x)
∆x+
The Product Rule
f ′(x) = lim∆x→0
u(x + ∆x)v(x + ∆x) − u(x)v(x)
∆x
Now we apply an algebraic trick: add and subtract u(x)v(x + ∆x).
u(x + ∆x)v(x + ∆x) − u(x)v(x + ∆x) + u(x)v(x + ∆x) − u(x)v(x)
∆x
= lim∆x→0
v(x + ∆x) [u(x + ∆x) − u(x)] + u(x) [v(x + ∆x) − v(x)]
∆x
= lim∆x→0
v(x+∆x)u(x + ∆x) − u(x)
∆x+ lim
∆x→0u(x)
v(x + ∆x) − v(x)
∆x
The Product Rule
The Product Rule
= lim∆x→0
v(x+∆x)u(x + ∆x) − u(x)
∆x+ lim
∆x→0u(x)
v(x + ∆x) − v(x)
∆x
The Product Rule
= lim∆x→0
v(x+∆x)u(x + ∆x) − u(x)
∆x+ lim
∆x→0u(x)
v(x + ∆x) − v(x)
∆x
= lim∆x→0
v(x + ∆x). lim∆x→0
u(x + ∆x) − u(x)
∆x+
The Product Rule
= lim∆x→0
v(x+∆x)u(x + ∆x) − u(x)
∆x+ lim
∆x→0u(x)
v(x + ∆x) − v(x)
∆x
=��
������
�:v(x)lim
∆x→0v(x + ∆x). lim
∆x→0
u(x + ∆x) − u(x)
∆x+
The Product Rule
= lim∆x→0
v(x+∆x)u(x + ∆x) − u(x)
∆x+ lim
∆x→0u(x)
v(x + ∆x) − v(x)
∆x
= lim∆x→0
v(x + ∆x).���
�����
����:u′(x)
lim∆x→0
u(x + ∆x) − u(x)
∆x+
The Product Rule
= lim∆x→0
v(x+∆x)u(x + ∆x) − u(x)
∆x+ lim
∆x→0u(x)
v(x + ∆x) − v(x)
∆x
= lim∆x→0
v(x + ∆x).���
�����
����:u′(x)
lim∆x→0
u(x + ∆x) − u(x)
∆x+
+u(x) lim∆x→0
v(x + ∆x) − v(x)
∆x
The Product Rule
= lim∆x→0
v(x+∆x)u(x + ∆x) − u(x)
∆x+ lim
∆x→0u(x)
v(x + ∆x) − v(x)
∆x
= lim∆x→0
v(x + ∆x).���
�����
����:u′(x)
lim∆x→0
u(x + ∆x) − u(x)
∆x+
+u(x)���
������
���:v ′(x)
lim∆x→0
v(x + ∆x) − v(x)
∆x
The Product Rule
= lim∆x→0
v(x+∆x)u(x + ∆x) − u(x)
∆x+ lim
∆x→0u(x)
v(x + ∆x) − v(x)
∆x
= lim∆x→0
v(x + ∆x).���
�����
����:u′(x)
lim∆x→0
u(x + ∆x) − u(x)
∆x+
+u(x)���
������
���:v ′(x)
lim∆x→0
v(x + ∆x) − v(x)
∆x
= v(x)u′(x) + u(x)v ′(x)
The Product Rule
The Product Rule
That is:
The Product Rule
That is:
(u(x)v(x))′ = u′(x)v(x) + u(x)v ′(x)
The Product Rule
That is:
(u(x)v(x))′ = u′(x)v(x) + u(x)v ′(x)
The Product Rule
That is:
(u(x)v(x))′ = u′(x)v(x) + u(x)v ′(x)
This is the product rule!