Differential Calculus
Rules of differentiation, tangent lines, optimization
Rules of Differentiation
Key rules:
• Power rule: d/dx[xⁿ] = nxⁿ⁻¹
• Constant rule: d/dx[c] = 0
• Sum rule: d/dx[f + g] = f' + g'
• Constant multiple: d/dx[cf] = c·f'
Applications include tangent lines, rates of change, and optimisation.
Example
Applying Differentiation Rules
f(x) = 3x⁴ − 2x³ + 5x − 7
f'(x) = 12x³ − 6x² + 5
Find the equation of the tangent to f(x) = x² at x = 3:
f(3) = 9, f'(x) = 2x, f'(3) = 6
Tangent: y − 9 = 6(x − 3) → y = 6x − 9
Optimisation (Maxima and Minima)
To find turning points: set f'(x) = 0 and solve. Use the second derivative to classify: f''(x) > 0 → minimum, f''(x) < 0 → maximum. Optimisation problems involve maximising profit, minimising cost, etc.
Example
Optimisation Problem
A farmer has 60 m of fencing. Find the maximum area of a rectangular camp.
Perimeter: 2x + 2y = 60 → y = 30 − x
Area: A = x(30−x) = 30x − x²
A'(x) = 30 − 2x = 0 → x = 15
y = 30 − 15 = 15
Maximum area = 15 × 15 = 225 m²
Note
Remember
The derivative gives the rate of change. f'(x) = 0 at turning points. Check endpoints for optimisation on closed intervals. The second derivative test confirms max/min. Cubic graphs have at most 2 turning points.
Key Vocabulary
DifferentiationThe process of finding the derivative of a function
TangentA line that touches a curve at one point with the same gradient
Turning pointA point where the graph changes direction (max or min)
OptimisationFinding the maximum or minimum value in a practical problem
Rate of changeHow fast a quantity is changing
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Differentiation
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