Bring the power down, subtract one: To differentiate xⁿ: multiply by the power, then reduce the power by 1 → n·xⁿ⁻¹. A constant differentiates to 0, and a constant multiple stays: d/dx(a·xⁿ) = a·n·xⁿ⁻¹.
IB-style question — single terms
Differentiate x⁵, 3x², and 7.
Step by step
- Power rule on x⁵ and 3x².
- A constant has gradient 0.
Final answer
5x⁴, 6x, and 0.
Constants vanish: Any standalone number differentiates to 0 — its graph is a flat line with zero gradient.
Differentiate term by term: For a polynomial, apply the power rule to each term separately and keep the + and − signs. The derivative of a sum is the sum of the derivatives.
IB-style question — a polynomial
Differentiate f(x) = 2x³ − 5x² + 4x − 9.
Step by step
- Power rule on each term.
- Combine (the −9 → 0).
Final answer
f'(x) = 6x² − 10x + 4.
4x → 4, not 4x: A term like 4x is 4x¹; its derivative is 4·1·x⁰ = 4 (the x disappears).
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Rewrite 1/xⁿ as a negative power first: Before differentiating a fraction like 1/xⁿ, rewrite it as x⁻ⁿ, then apply the power rule. Remember subtracting 1 from a negative power makes it more negative.
IB-style question — a negative power
Differentiate f(x) = 3/x².
Step by step
- Rewrite as a power.
- Power rule: −2 down, power becomes −3.
Final answer
f'(x) = −6x⁻³ = −6/x³.
−2 − 1 = −3: Reducing the power means subtracting 1: −2 → −3 (not −1). A sign slip here is the classic mistake.
Write roots as fractional powers first: A root like √x is x^(1/2); differentiate with the power rule, then rewrite the answer with a root if you like. (√x → ½x−1/2 = 1/(2√x).)
IB-style question — a root
Differentiate f(x) = 4x3/2, then differentiate g(x) = √x.
Step by step
- Power rule on 4x3/2.
- And √x = x1/2.
Final answer
f'(x) = 6√x; g'(x) = 1/(2√x).
Subtract 1 from the fraction: 3/2 − 1 = 1/2 and 1/2 − 1 = −1/2 — keep the fractions, then tidy into root form at the end.