Power, Product, and Quotient Rules
Shortcut rules let you differentiate without going back to the limit definition every time: the power rule for x^n, the product rule for products of functions, and the quotient rule for ratios.
Power rule and basic combinations
Power rule: d/dx[xⁿ] = n·x^(n-1), valid for any real exponent n. Constant multiple rule: d/dx[k·f(x)] = k·f'(x). Sum/difference rule: derivatives distribute over + and -.
Product rule: d/dx[f(x)·g(x)] = f'(x)g(x) + f(x)g'(x). A common mistake is assuming the derivative of a product is just the product of the derivatives — it is NOT; you must use the full product rule.
Quotient rule: d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]². Order matters in the numerator — swapping f and g's roles gives the wrong sign.
Worked Example
Differentiate h(x) = x²·sin(x).
- Identify f(x)=x² and g(x)=sin(x), so f'(x)=2x and g'(x)=cos(x).
- Apply the product rule: h'(x) = f'(x)g(x) + f(x)g'(x) = 2x·sin(x) + x²·cos(x).
Answer: h'(x) = 2x·sin(x) + x²·cos(x)
Key terms (3)
- Power rule
- d/dx[xⁿ] = n·x^(n-1)
- Product rule
- d/dx[fg] = f'g + fg'
- Quotient rule
- d/dx[f/g] = (f'g - fg') / g²
Practice Quiz
Differentiate f(x) = 4x³ - 2x + 7.