finding f'(x) and the square root of f'(2) [differentiation of rational functions]

After watching this video, you would be able to differentiate the rational function f(x) and compute the square root of f'(2). That is; finding f'(x) and computing the square root of f'(2). Differentiation of Rational Functions To differentiate a rational function: 1. *Quotient Rule*: If f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x)) / h(x)^2 Example Find the derivative of f(x) = (2x + 1) / (x + 2) 1. *Identify g(x) and h(x)*: g(x) = 2x + 1, h(x) = x + 2 2. *Find g'(x) and h'(x)*: g'(x) = 2, h'(x) = 1 3. *Apply Quotient Rule*: f'(x) = ((x + 2)(2) - (2x + 1)(1)) / (x + 2)^2 4. *Simplify*: f'(x) = (2x + 4 - 2x - 1) / (x + 2)^2 = 3 / (x + 2)^2 Computing the Square Root of f'(2) Given f(x) = (2x + 1) / (x + 2), we previously found f'(x) = 3 / (x + 2)^2. Step 1: Find f'(2) 1. *Substitute x = 2*: f'(2) = 3 / (2 + 2)^2 = 3 / 4^2 = 3 / 16 Step 2: Compute the Square Root 1. *Find the square root*: √(f'(2)) = √(3 / 16) = √3 / √16 = √3 / 4 Tips 1. *Simplify before differentiating*: Simplify the rational function before applying the Quotient Rule. 2. *Use the Quotient Rule*: Apply the Quotient Rule when differentiating rational functions. #calculus #differentiation #differentiating #differentiationtricks #quotientrule #rationalfunctions #differentiationformulas #square #squareroot #squareroottricks #derivatives #derivatives #maths #finding #solving #mathematics #mathematician #mathematicseducation #olympiad #olympiadmathematics #usa #uk #canada #china #nigeria #india #indianmathematics