Algebraic and Trigonometric Differentiation
Simplify to powers where needed and differentiate algebraic, sine and cosine terms with accurate coefficients.
How to study Higher Mathematics
Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.
Core concepts
Concept 1
power rule
Exam cue: Can the function first be rewritten as powers of x?
Concept 2
negative and fractional powers
Exam cue: Does the inner coefficient multiply the trig derivative?
Concept 3
sin ax derivative
Exam cue: Is the current line y, f(x), dy/dx or f′(x)?
Concept 4
cos ax derivative
Concept 5
derivative notation
Risk pitfalls and guardrails
Differentiating before simplifying
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Omitting the coefficient a in sin(ax) or cos(ax)
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Labelling a derivative line as y
Guardrail: Use correct notation and check coefficients, signs and the constant of integration.
Memory anchors
Power down, power down one
For xⁿ, multiply by n and reduce the exponent by one.
Rewrite roots and reciprocals
Power notation makes the rule visible.
Sine becomes cosine
d/dx sin(ax) = a cos(ax).
Cosine becomes negative sine
d/dx cos(ax) = −a sin(ax).
Constants vanish
A constant term has derivative zero.
Notation tells the object
Keep the original function distinct from its derivative.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Differentiate y=7x^4.
Differentiate y=3x^5−2x²+6.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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