Differentiation

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Chain Rule Formula: If y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx) Alternatively: If y = f[g(x)], then dy/dx = f'[g(x)] × g'(x) Use when: Differentiating functions within functions, like s…

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Step 1: Identify functions - Outer function: √u where u = x² + 5 - Inner function: u = x² + 5 Step 2: Find derivatives - dy/du = 1/(2√u) = 1/(2√(x² + 5)) - du/dx = 2x Step 3: Apply chain rule dy/dx …

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Inverse Function Theorem: If y = f(x) is differentiable and dy/dx ≠ 0, and x = f⁻¹(y) exists, then: dx/dy = 1/(dy/dx) Geometric Meaning: - The slope of f⁻¹ at point (a,b) is the reciprocal of the sl…

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Given: y = sin⁻¹(x), where -1 ≤ x ≤ 1, -π/2 ≤ y ≤ π/2 Step 1: Express inverse relationship If y = sin⁻¹(x), then x = sin y Step 2: Differentiate x = sin y with respect to y dx/dy = cos y Step 3: Ap…

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1. d/dx[sin⁻¹(x)] = 1/√(1-x²), |x| < 1 2. d/dx[cos⁻¹(x)] = -1/√(1-x²), |x| < 1 3. d/dx[tan⁻¹(x)] = 1/(1+x²), x ∈ ℝ 4. d/dx[cot⁻¹(x)] = -1/(1+x²), x ∈ ℝ 5. d/dx[sec⁻¹(x)] = 1/(x√(x²-1)), |x| > 1 …

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Method 1: Direct differentiation dy/dx = 1/(1+(2x/(1-x²))²) × d/dx[2x/(1-x²)] Using quotient rule for 2x/(1-x²): d/dx[2x/(1-x²)] = [2(1-x²) - 2x(-2x)]/(1-x²)² = (2+2x²)/(1-x²)² Therefore: dy/dx = 2/…

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Use logarithmic differentiation when: 1. Function has form y = [f(x)]^g(x) (variable base and exponent) 2. Products/quotients with multiple terms and powers 3. Complex expressions with roots, powers, …

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Step 1: Take natural log of both sides ln y = ln(x^x) = x ln x Step 2: Differentiate both sides with respect to x d/dx(ln y) = d/dx(x ln x) Step 3: Left side using chain rule (1/y)(dy/dx) = d/dx(x l…