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Calculus
· Formula🧮 Formula Reference Sheet
Derivative: d/dx (xⁿ) = nxⁿ⁻¹ Product rule: (uv)' = u'v + uv' Chain rule: (f(g(x)))' = f'(g(x))·g'(x) Integral: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C Fundamental thm: ∫ₐᵇ f'(x)dx = f(b) − f(a)
⚡ Example per formula
Derivative: d/dx (xⁿ) = nxⁿ⁻¹
↳d/dx(x⁵) = 5x⁴
Product rule: (uv)' = u'v + uv'
↳d(x²·sin x) = 2x sin x + x² cos x
Chain rule: (f(g(x)))' = f'(g(x))·g'(x)
↳d(sin(x²)) = 2x cos(x²)
Integral: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
↳∫ x² dx = x³/3 + C
Fundamental thm: ∫ₐᵇ f'(x)dx = f(b) − f(a)
↳∫₀¹ 2x dx = [x²]₀¹ = 1
✏️ Worked Example
Differentiate f(x) = 4x³ − 2x + 7
👁 Show step-by-step solution
f'(x) = 4·3x² − 2 + 0 = 12x² − 2. ✅ Answer: 12x² − 2
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