Differentiation

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The derivative of a function f(x) at point x=a is defined as: f'(a) = lim[h→0] (f(a+h) - f(a))/h This represents the rate of change of the function at that point. The process of finding this limit is

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Step 1: f(x) = x³, so f(x+h) = (x+h)³ Step 2: Apply first principle formula: f'(x) = lim[h→0] [(x+h)³ - x³]/h Step 3: Expand (x+h)³: (x+h)³ = x³ + 3x²h + 3xh² + h³ Step 4: Substitute: f'(x) = lim[h

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Power Rule: If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹ (where n is any real number) Example: Find derivative of f(x) = x⁵ Solution: Using power rule: f'(x) = 5x⁵⁻¹ = 5x⁴ Another example: f(x) = x⁻² = 1/x² f'(

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Step 1: f(x) = √x, f(x+h) = √(x+h) Step 2: Apply first principle: f'(x) = lim[h→0] [√(x+h) - √x]/h Step 3: Rationalize by multiplying by conjugate: f'(x) = lim[h→0] [√(x+h) - √x]/h × [√(x+h) + √x]/[

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Standard Derivatives: 1. d/dx(c) = 0 (constant rule) 2. d/dx(x^n) = nx^(n-1) (power rule) 3. d/dx(e^x) = e^x 4. d/dx(a^x) = a^x ln(a) (where a > 0) 5. d/dx(ln x) = 1/x 6. d/dx(log_a x) = 1/(x ln a) 7

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Sum Rule: d/dx[u + v] = du/dx + dv/dx Difference Rule: d/dx[u - v] = du/dx - dv/dx Example: Find derivative of f(x) = 3x⁴ - 5x² + 7x - 2 Step 1: Apply sum/difference rule: f'(x) = d/dx(3x⁴) - d/dx(5

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Product Rule: If y = u·v, then dy/dx = u(dv/dx) + v(du/dx) Example: Find d/dx[x²(3x + 1)] Step 1: Identify u and v: u = x², v = 3x + 1 Step 2: Find derivatives: du/dx = 2x dv/dx = 3 Step 3: Apply

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Quotient Rule: If y = u/v, then dy/dx = [v(du/dx) - u(dv/dx)]/v² Example: Find d/dx[(2x + 1)/(x² - 1)] Step 1: Identify u and v: u = 2x + 1, v = x² - 1 Step 2: Find derivatives: du/dx = 2, dv/dx =