Differentiation

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The derivative of f(x) from first principles is: f'(x) = lim[h→0] [f(x+h) - f(x)]/h This represents the instantaneous rate of change of f(x) with respect to x. The process of finding this derivative

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Geometric interpretation: dy/dx represents the slope of the tangent line to the curve at point P. Step-by-step understanding: 1. Consider points P(x,y) and Q(x+δx, y+δy) on the curve 2. Slope of seca

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Power Rule: d/dx(xⁿ) = nxⁿ⁻¹ Proof using first principles: Let y = xⁿ y + δy = (x + δx)ⁿ δy = (x + δx)ⁿ - xⁿ = xⁿ[(1 + δx/x)ⁿ - 1] Using binomial expansion: (1 + δx/x)ⁿ = 1 + n(δx/x) + n(n-1)/2!(

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Step-by-step solution: Given: S = 3t² Find: Velocity at t = 4 seconds Step 1: Velocity = dS/dt (derivative of distance) Step 2: Apply power rule to S = 3t² dS/dt = 3 × d/dt(t²) = 3 × 2t = 6t

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Derivative of constant: d/dx(c) = 0 Proof: Let y = c (constant) y + δy = c (since c doesn't change with x) δy = c - c = 0 δy/δx = 0/δx = 0 dy/dx = lim[δx→0] (0) = 0 Alternatively using power rule: c

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Sum Rule: d/dx[f(x) + g(x)] = d/dx[f(x)] + d/dx[g(x)] Difference Rule: d/dx[f(x) - g(x)] = d/dx[f(x)] - d/dx[g(x)] Step-by-step example: Find d/dx(x³ + 2x² - 5x + 7) Step 1: Apply sum/difference rul

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Step-by-step solution: Step 1: Rewrite using negative powers y = x³ + x⁻² - x⁻¹ Step 2: Apply sum/difference rule dy/dx = d/dx(x³) + d/dx(x⁻²) - d/dx(x⁻¹) Step 3: Apply power rule to each term • d/

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Product Rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) Or: d/dx(uv) = u'v + uv' Solution for d/dx[(x² + 1)(3x - 2)]: Step 1: Identify functions f(x) = x² + 1, g(x) = 3x - 2 Step 2: Find derivatives f