Differentiation — Flashcards
Maharashtra Board · Class 12 · Mathematics & Statistics -Commerce
20 flashcards for Differentiation (Maharashtra Board Class 12 Mathematics & Statistics -Commerce) to test yourself on key terms and facts.
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What is the Chain Rule formula for differentiating composite functions?
Answer
If y = f(u) and u = g(x), then: dy/dx = (dy/du) × (du/dx) This is used when we have a function inside another function. Example: For y = (3x² + 5)⁴ Let u = 3x² + 5, so y = u⁴ dy/du = 4u³, du/dx = 6x…
Find dy/dx for y = (4x³ + 3x² - 2x)⁶
Answer
Step-by-step solution: Step 1: Identify the composite function Let u = 4x³ + 3x² - 2x, so y = u⁶ Step 2: Find derivatives dy/du = 6u⁵ du/dx = 12x² + 6x - 2 Step 3: Apply chain rule dy/dx = (dy/du) ×…
What is the formula for derivative of inverse functions?
Answer
If y = f(x) has an inverse function x = f⁻¹(y), then: dx/dy = 1/(dy/dx), provided dy/dx ≠ 0 This means the derivative of the inverse function is the reciprocal of the derivative of the original funct…
Find the rate of change of demand (x) with respect to price (y) if y = 20 + 15x + x²
Answer
Step-by-step solution: Step 1: Find dy/dx dy/dx = 15 + 2x Step 2: Apply inverse function rule dx/dy = 1/(dy/dx) dx/dy = 1/(15 + 2x) Step 3: Interpret Rate of change of demand with respect to price =…
What is logarithmic differentiation and when is it used?
Answer
Logarithmic differentiation is used for functions involving: 1. Products of multiple functions 2. Quotients of complex functions 3. Functions with variable exponents [f(x)]^g(x) Method: 1. Take natur…
Find dy/dx for y = (3 + x)^x using logarithmic differentiation
Answer
Step-by-step solution: Step 1: Take natural logarithm ln y = ln[(3 + x)^x] = x ln(3 + x) Step 2: Differentiate both sides (1/y) × (dy/dx) = x × 1/(3 + x) + ln(3 + x) × 1 (1/y) × (dy/dx) = x/(3 + x) +…
Find dy/dx for y = [(2x + 3)⁵]/[(3x - 1)³√(5x - 2)]
Answer
Step-by-step solution: Step 1: Rewrite using negative exponents y = (2x + 3)⁵(3x - 1)⁻³(5x - 2)⁻¹/² Step 2: Take natural logarithm ln y = 5ln(2x + 3) - 3ln(3x - 1) - (1/2)ln(5x - 2) Step 3: Differen…
What is an implicit function and how do we differentiate it?
Answer
Implicit Function: A relation where y cannot be easily expressed as y = f(x) Examples: x² + y² = 25, xy = log(xy) Differentiation Process: 1. Differentiate both sides with respect to x 2. Use chain r…
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