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UniversityCalculus
Differentiation (Calculus)
Finding rates of change and gradients of curves
Learn differential calculus — finding derivatives to calculate rates of change and optimise functions.
✓ Power rule: d/dx(xⁿ) = nxⁿ⁻¹✓ Constant rule: d/dx(c) = 0✓ d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x
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📖 Understanding Differentiation (Calculus)
Differentiation is the process of finding the derivative of a function. The derivative dy/dx (or f'(x)) represents the instantaneous rate of change of y with respect to x — the gradient of the curve at any point.
Basic rule (Power Rule): If y = xⁿ, then dy/dx = nxⁿ⁻¹. Bring the power down and reduce the power by 1.
Key derivatives: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x.
Chain Rule: If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). Product Rule: d/dx(uv) = u'v + uv'. Quotient Rule: d/dx(u/v) = (u'v − uv')/v².
🔑 Key Points to Remember
- ✓Power rule: d/dx(xⁿ) = nxⁿ⁻¹
- ✓Constant rule: d/dx(c) = 0
- ✓d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x
- ✓Chain rule: dy/dx = dy/du × du/dx
- ✓Set dy/dx = 0 to find stationary points
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