12. Product Rule, Quotient Rule, Chain Rule
Remember these rules for the rest of your life.
Product Rule
Quotient Rule
Original Function
Derivative
h(x) = f (x) · g(x)
h0 (x) = f (x) · g 0 (x) + g(x) · f 0 (x)
h(x) =
f (x)
g(x)
h(x) = f g(x)
Chain Rule
√
1. Find y 0 for y = x3 x−2 + 4x. What rule(s) do you use?
2. Differentiate h(x) = (x5 − 3x)2 (x − x−1 ). What rule(s) do you use?
3. Find
ds
dt
for s(t) =
4. Find
d
dx
h
2x−x−4
x2 +x+1
i
t2 −1
t3 −1 .
What rule(s) do you use?
. What rule(s) do you use?
1
h0 (x) =
g(x)·f 0 (x)−f (x)·g 0 (x)
[g(x)]2
h0 (x) = f 0 g(x) · g 0 (x)
5. Differentiate y = f g(x) where f (x) = x − 3 and g(x) =
6. Find
d2 y
dx2
2
x+2 .
What rule(s) do you use?
√
for y = x2 3 x2 − 2x−1 .
7. Solve for h0 (x) in terms of x and f (x) when h(x) = (x3 − x3 ) · [f (x)]2 .
2