Question Detail

The 'first principle' method refers to finding the derivative of a function using its definition as a limit of the average rate of change.Given y = f(x) = √x.1. Find Δy: Δy = f(x + Δx) - f(x) = √(x + Δx) - √x.2. Find the average rate of change, Δy/Δx: Δy/Δx = (√(x + Δx) - √x) / Δx.The question asks for 'Δy/Δx' from first principles. While the expression itself is (√(x + Δx) - √x) / Δx, none of the options directly present this expression. However, the phrase 'from first principle find dy/dx' usually implies calculating the limit of Δy/Δx as Δx approaches 0.Option C, lim<sub>Δx→0</sub> Δy/Δx, is the formal definition of the derivative dy/dx using the first principle. Therefore, conceptually, C is the correct answer as it represents the process of finding the derivative from first principles.To complete the process and find the derivative:limΔx→0 (√(x + Δx) - √x) / Δx= limΔx→0 [(√(x + Δx) - √x) / Δx] * [(√(x + Δx) + √x) / (√(x + Δx) + √x)] (Multiply by the conjugate)= limΔx→0 [(x + Δx) - x] / [Δx(√(x + Δx) + √x)]= limΔx→0 Δx / [Δx(√(x + Δx) + √x)]= limΔx→0 1 / (√(x + Δx) + √x)= 1 / (√x + √x) = 1 / (2√x).While 1/(2√x) is the derivative, option C correctly states the definition for finding it from first principles.