The Chain Rule is one of the most important and widely used differentiation rules. It tells us how to find the derivative of a composite function (a function inside another function).
๐ The Formula
If y=f(g(x)), then:
๐ Practice Problems
1Problem 1medium
โ Question:
Find the derivative of f(x)=(2x3 using the Chain Rule.
๐ Worked Example: Related Rates โ Expanding Circle
Problem:
A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of 2 cm/s. How fast is the area of the circle increasing when the radius is 10 cm?
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this The Chain Rule study guide free?โพ
Yes โ all study notes, flashcards, and practice problems for The Chain Rule on Study Mondo are 100% free. No account is needed to access the content.
What course covers The Chain Rule?โพ
The Chain Rule is part of the AP Calculus AB course on Study Mondo, specifically in the Derivatives section. You can explore the full course for more related topics and practice resources.
Are there practice problems for The Chain Rule?
dxdyโ=fโฒ(g(x))โ gโฒ(x)
Or in Leibniz notation:
dxdyโ=dudyโโ dxduโ
where u=g(x).
๐ก Memory Trick: "Derivative of the outside times derivative of the inside"
Understanding Composite Functions
A composite function is a function within another function.
Examples of Composite Functions
f(x)=(x2+1)5 โ The function x2+1 is inside the power of 5
f(x)=sin(3x) โ The function 3x is inside the sine function
f(x)=ex2 โ The function x2 is inside the exponential
f(x)=x3โ2xโ โ The function is inside the square root
The "Outside-Inside" Method
Step-by-Step Process
Identify the outside function and the inside function
Differentiate the outside function (leave the inside alone)
Multiply by the derivative of the inside function
Example Walkthrough
Find the derivative of f(x)=(x2+3x)4
Step 1: Identify
Outside function: u4 (something to the 4th power)
Inside function: u=x2+3x
Step 2: Differentiate the outside
Derivative of u4 is 4u3
Replace u with (x2+3x): 4(x2+3x)3
Step 3: Multiply by derivative of inside
Derivative of (x2+3x) is 2x+3
Final answer: fโฒ(x)=4(x2+3x)3(2x+3)
Common Chain Rule Patterns
Pattern 1: Powers of Functions
If f(x)=[g(x)]n, then fโฒ(x)=n[g(x)]nโ1โ gโฒ(x)
While (3x2โ1)6โ 6x is correct, it's better to write 6x(3x2โ1)6 (constants first).
Mistake 3: Confusing Inside and Outside
Make sure you identify which function is inside and which is outside!
๐ฏ When to Use the Chain Rule
Use the Chain Rule whenever you see:
A function raised to a power (other than just x)
Trig functions with anything other than just x inside
Exponentials with anything other than just x in the exponent
Square roots or other roots of expressions
Any function within another function
๐ก Quick Test: If you can't apply the Power Rule, Product Rule, or Quotient Rule directly, you probably need the Chain Rule!
Multiple Compositions
Sometimes you need to apply the Chain Rule more than once!
Example: f(x)=sin2(3x)=[sin(3x)]2
This has TWO layers:
Outside: squaring function
Middle: sine function
Inside: 3x
Solution:
Derivative of square: 2sin(3x)
Times derivative of sine: 2sin(3x)โ cos(3x)
Times derivative of 3x: 2sin(3x)cos(3x)โ 3
Answer: fโฒ(x)=6sin(3x)cos(3x)
๐ Practice Strategy
Circle the inside function
Box the outside function
Write "outside' ร inside'" as a reminder
Apply the formula step by step
Simplify your answer
โ
5x+
1)6
๐ก Show Solution
Step 1: Identify the functions
Outside function: u6 (something to the 6th power)
Inside function: u=2x3โ5x+1
Step 2: Apply the Chain Rule
Using dxdโ[u6]=6u
Step 3: Find the derivative of the inside
dxduโ=dx
Step 4: Put it together
fโฒ(x)=6(2x3โ5x+
This can also be written as:
fโฒ(x)=6(6x2โ5)(2x
Answer: fโฒ(x)=6(6x2โ5)(2
2Problem 2medium
โ Question:
Find the derivative of each function:
a) f(x)=(3x2โ5)7
b) g(x)=sin(4x)
c) h(x)=ex2+1
๐ก Show Solution
Solution:
Part (a): Chain rule: dxdโ
3Problem 3medium
โ Question:
Find dxdyโ if y=4x2+9โ.
๐ก Show Solution
Step 1: Rewrite using exponents
y=4x2+9
4Problem 4hard
โ Question:
Find dxdyโ if y=cos(3x2)โ.
๐ก Show Solution
Solution:
Rewrite: y=[cos(3x2)]1/2
This requires chain rule twice (nested composition).
Let , then
5Problem 5hard
โ Question:
Find the derivative of g(x)=(x2+1)3(2xโ5)4. (This requires both Product Rule AND Chain Rule!)
๐ก Show Solution
This problem requires both the Product Rule and the Chain Rule.
Step 1: Identify that this is a product
Let u=(x2+1)3 and
The Power Rule
โพ
Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.