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The derivative from first principles is defined as f'(x) = lim (Δx→0) [f(x+Δx) - f(x)] / Δx.Here, Δy/Δx = [f(x+Δx) - f(x)] / Δx.Given f(x) = √x, then f(x+Δx) = √(x+Δx).Δy/Δx = [√(x+Δx) - √x] / ΔxTo evaluate the limit, multiply the numerator and denominator by the conjugate of the numerator, √(x+Δx) + √x:Δy/Δx = ([√(x+Δx) - √x] / Δx) * ([√(x+Δx) + √x] / [√(x+Δx) + √x])= [(x+Δx) - x] / [Δx * (√(x+Δx) + √x)]= Δx / [Δx * (√(x+Δx) + √x)]= 1 / [√(x+Δx) + √x]Now, take the limit as Δx→0:lim (Δx→0) [1 / (√(x+Δx) + √x)] = 1 / (√x + √x) = 1 / (2√x)