The basic derivatives to know: Alongside the power rule, learn these: sin x → cos x, cos x → −sin x, eˣ → eˣ (unchanged!), and ln x → 1/x. Differentiate sums and multiples term by term, as before.
IB-style question — a mix
Differentiate y = 3 sin x + 2eˣ − ln x.
Step by step
- Differentiate each term using the standard derivatives.
Final answer
dy/dx = 3cos x + 2eˣ − 1/x.
cos x picks up a minus: d/dx(cos x) = −sin x — the negative sign is the classic slip. And eˣ stays eˣ.
Differentiate the outside, times the inside's derivative: For a composite y = f(g(x)) (a 'function of a function'), the chain rule says dy/dx = f'(g(x)) · g'(x) — differentiate the outer function, then multiply by the derivative of the inside.
IB-style question — a composite power
Differentiate y = (x² + 3)⁴.
Step by step
- Outer is ( )⁴, inner is x² + 3. Differentiate the outer.
- Multiply by the inner's derivative (2x).
Final answer
dy/dx = 8x(x² + 3)³.
Spot the inside: Identify the inside function (what's in the bracket / argument); its derivative is the extra factor you multiply by.
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n(bracket)ⁿ⁻¹ × (bracket)': For (ax + b)ⁿ, the chain rule gives n(ax + b)ⁿ⁻¹ × a — bring the power down, reduce it by 1, and multiply by the bracket's derivative.
IB-style question — bracket to a power
Differentiate y = (3x − 1)⁶.
Step by step
- Power down, reduce power, keep the bracket.
- Multiply by the derivative of the inside (3).
Final answer
dy/dx = 18(3x − 1)⁵.
Don't forget the inner derivative: Leaving off the × (bracket)' factor is the most common chain-rule error (here the × 3).
Same idea: outer derivative × inner derivative: For composites of trig/exp/log: sin(ax+b) → a cos(ax+b), e^(ax+b) → a·e^(ax+b), ln(ax+b) → a/(ax+b). The inner derivative a appears as a multiplier.
IB-style question — trig & exponential
Differentiate y = sin(3x) and y = e4x and y = ln(2x + 1).
Step by step
- sin(3x): cos of it, times the inner derivative 3.
- e4x and ln(2x+1).
Final answer
3cos(3x); 4eˣ form 4e4x; and 2/(2x + 1).
The multiplier is the inner derivative: The number out front is always the derivative of the inside — e.g. the 3 in sin(3x), the 2 in ln(2x+1).