Differentiation rules are not four unrelated formulas. Each rule answers a structural question: is the function a power, a product, a quotient, or a function inside another function?
The central skill is therefore recognition before calculation. A student who identifies the structure correctly can usually choose the rule quickly. A student who begins differentiating term-by-term without reading the structure may produce a polished answer to the wrong problem.
1. Power rule
For y=x^n, the derivative is dy/dx=nx^(n−1), for the powers used in this course wherever the function is differentiable.
Examples: d/dx(x⁵)=5x⁴; d/dx(3x⁴)=12x³; d/dx(7)=0.
2. Rewrite before differentiating
Expressions containing roots or reciprocals can often be rewritten as powers. √x=x^(1/2), so d/dx(√x)=1/(2√x) for x>0. Likewise 1/x=x^−1, giving derivative −x^−2=−1/x² for x≠0.
Rewriting can simplify rule selection and make sign errors easier to detect.
3. Product rule
If y=u(x)v(x), then
dy/dx=u′v+uv′.
Differentiate one factor at a time while leaving the other unchanged.
4. Worked product example
Differentiate y=(x²+1)(3x−4).
Let u=x²+1 and v=3x−4. Then u′=2x and v′=3.
dy/dx=2x(3x−4)+3(x²+1)=6x²−8x+3x²+3=9x²−8x+3.
Independent check: expand first, y=3x³−4x²+3x−4, then differentiate to obtain 9x²−8x+3.
5. Quotient rule
If y=u/v, then
dy/dx=(u′v−uv′)/v², where v≠0.
The order in the numerator matters. A reliable verbal cue is “differentiate the top times the bottom, minus the top times differentiate the bottom.”
6. Worked quotient example
Differentiate y=(x²+3)/(x−1).
u=x²+3, u′=2x; v=x−1, v′=1.
dy/dx=[2x(x−1)−(x²+3)]/(x−1)².
Simplify the numerator: 2x²−2x−x²−3=x²−2x−3. Hence
dy/dx=(x²−2x−3)/(x−1)², x≠1.
7. Chain rule
If y=f(g(x)), then
dy/dx=f′(g(x))g′(x).
Differentiate the outside function while keeping the inside expression, then multiply by the derivative of the inside.
8. Worked chain example
Differentiate y=(3x²−1)^5.
Outside: fifth power. Inside: 3x²−1.
dy/dx=5(3x²−1)^4(6x)=30x(3x²−1)^4.
A common incomplete answer is 5(3x²−1)^4, which differentiates only the outer layer.
9. Product plus chain rule
Differentiate y=x²(2x+1)^3.
The whole function is a product, and the second factor itself needs the chain rule.
dy/dx=2x(2x+1)^3+x²·3(2x+1)²·2.
Factor if useful:
dy/dx=2x(2x+1)²[(2x+1)+3x]=2x(2x+1)²(5x+1).
10. Quotient plus chain rule
For y=(x+1)/(x²+4)^2, the denominator is a composite function. You may use the quotient rule directly, differentiating the denominator by chain rule, or rewrite as (x+1)(x²+4)^−2 and use product plus chain rule.
Equivalent routes should agree. Choosing the shorter route is a method-selection decision, not a different mathematics.
11. Simplify before or after?
If y=(x²−1)/(x−1), then for x≠1 the expression simplifies to x+1. Differentiating the simplified form gives 1 on that domain.
The original function is still undefined at x=1. Algebraic simplification does not silently restore an excluded point.
12. Rule-selection tree
- Can the expression be simplified first?
- Is it a sum of separate terms? Differentiate term by term.
- Is the whole structure a product? Consider product rule.
- Is it a quotient? Consider quotient rule or negative powers.
- Is there a function inside another function? Use chain rule.
13. Common mistakes
- Using (uv)′=u′v′. This is generally false.
- Reversing the subtraction in the quotient rule.
- Forgetting the inner derivative in a composite function.
- Differentiating a constant as 1 instead of 0.
- Expanding a complicated product when factored differentiation would be shorter.
- Dropping domain restrictions after algebraic cancellation.
14. Diagnostic counterexample
Why is (x·x)′ not 1·1=1? Because x·x=x², whose derivative is 2x. Product rule gives 1·x+x·1=2x. A single simple counterexample is enough to disprove the false product rule.
15. Independent practice
- Differentiate 7x⁵−3x²+4.
- Differentiate (x²+2)(x³−1).
- Differentiate (2x+5)/(x−3).
- Differentiate (4x−1)^6.
- Differentiate x(3x²+1)^4.
- Differentiate (x²+1)^−3.
16. Answers
- 35x⁴−6x.
- 2x(x³−1)+3x²(x²+2).
- [2(x−3)−(2x+5)]/(x−3)²=−11/(x−3)².
- 24(4x−1)^5.
- (3x²+1)^4+24x²(3x²+1)^3.
- −6x(x²+1)^−4.
17. What mastery looks like
Rule mastery means more than recalling formulas. A strong student can identify the outer structure, simplify before differentiating when useful, combine rules correctly, preserve restrictions and verify equivalent derivative forms.
Continue with What Differentiation Means, the Differentiation guide, or return to the Additional Mathematics Hub.