The power rule: the derivative of xⁿ is n·x^(n−1). Deriving x² by hand gave 2x, which fits.
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, because subtraction is not commutative.
The chain rule
For a function inside a function, differentiate the outside, then multiply by the derivative of the inside.
The power rule
The derivative of xⁿ is nxⁿ⁻¹, for every real n. It can be proved from the difference quotient using the binomial theorem, and it handles polynomials, roots and reciprocals once fractional and negative exponents are allowed.
Sums and constant multiples
The derivative of a sum is the sum of the derivatives, and a constant factor passes straight through. Differentiation is linear, which is why polynomials can be differentiated term by term.
The standard derivatives
sin x gives cos x, cos x gives −sin x, eˣ gives eˣ, and ln x gives 1/x. These are proved from the definition once and then used freely. The exponential being its own derivative is what makes e the natural base.
The rules are shortcuts, not magic
Every rule is a theorem proved from the difference quotient. Working one out from the definition once — the power rule for x², say — makes the whole toolkit feel earned rather than handed down.
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 Priya Use the Chain Rule
Priya differentiates y = (3x + 1)⁴.
- Step 1
The outside function is something to the fourth power.
Step 4: Your Turn
Practice makes it stick.
The Power Rule
Problem 1 of 2
f(x) = x⁵. What is the coefficient in f′(x) = ?x⁴?
The Value
Problem 2 of 2
f(x) = x³, so f′(x) = 3x². What is f′(2)?
Apply the Rules
1 of 4
f(x) = x⁴. Coefficient in f′(x) = ?x³?
2 of 4
f(x) = x³, f′(x) = 3x². What is f′(3)?
3 of 4
f(x) = 7x. What is f′(x)?
4 of 4
Is the derivative of a product the product of the derivatives?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f(x) = x⁶. Coefficient in f′(x) = ?x⁵?
Question 2 of 2
What does the chain rule handle?
What You Learned
- The power rule drops the exponent and reduces it by one.
- Products and quotients have their own rules; they do not distribute.
- The chain rule multiplies by the derivative of the inside.