The power rule: the derivative of xⁿ is n·xⁿ⁻¹. Bring the exponent down, then reduce it by one.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The product rule
(fg)' = f'g + fg'. The derivative of a product is not the product of the derivatives.
The quotient rule
(f/g)' = (f'g − fg') / g². The order in the numerator matters: subtraction is not commutative.
Why the naive guess fails
Try f = g = x. The product is x², whose derivative is 2x, not 1 × 1.
The power rule and its reach
The derivative of xⁿ is nxⁿ⁻¹ for every real n, which covers polynomials, roots and reciprocals once fractional and negative exponents are allowed. Rewriting a root as a power before differentiating is usually the easiest route.
Products do not differentiate termwise
(uv)′ = u′v + uv′. The naive guess fails on x·x, where the true derivative is 2x rather than 1. Testing the guess against a case you can check is a good way to remember why the rule exists.
The quotient rule and its order
(u/v)′ = (u′v − uv′)/v². The subtraction means order matters, unlike in the product rule. Reversing the terms gives exactly the negative of the correct answer, which is easy to miss.
Simplify before differentiating
Many quotients are easier as products with negative exponents, and many products expand into simple polynomials. Choosing the easier route is legitimate and often halves the work.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, -2)
- Slope of the tangent0
Step 3: Watch an Example
One step at a time.
Watch Nadia Use the Product Rule
Nadia differentiates f(x) = x² · x³ two ways.
- Step 1
By the product rule: 2x · x³ + x² · 3x².
Step 4: Your Turn
Practice makes it stick.
The Power Rule
Problem 1 of 2
f(x) = x⁶. What is the coefficient in the derivative?
Evaluated
Problem 2 of 2
f(x) = x⁴. What is f'(2)?
Apply the Rules
1 of 8
f(x) = x⁵. f'(1)?
2 of 8
f(x) = 3x². f'(2)?
3 of 8
f(x) = x⁻¹. The derivative is −x⁻². What is f'(1)?
4 of 8
Match each rule to its formula.
Tap a card on the left to start.
5 of 8
f(x) = x² + x³. f'(1)?
6 of 8
Is (fg)' equal to f'g'? 1 for yes, 0 for no.
7 of 8
f(x) = 10x. f'(4)?
8 of 8
f(x) = x⁷. f'(1)?
Step 5: Quick Check
Show what you know.
Question 1 of 1
f(x) = x³. f'(3)?
What You Learned
- The power rule brings the exponent down and reduces it by one.
- The product rule is f'g + fg', not the product of derivatives.
- The quotient rule subtracts in a fixed order, over the denominator squared.