find the derivative of the following function

Question

𝑓(𝑥) = [(x² − 9) √𝑥 + 2]²

show solution

Answer 1

Solution:

let y = 𝑓(𝑥) = [(x² − 9) √𝑥 + 2]² Use chain rule to find f'(x)

y' = f'(x) = 2[(x² − 9) √𝑥 + 2]^2-1( d( (x² − 9) √𝑥 + 2) /dx) = 2[(x² − 9) √𝑥 + 2] [ d( (x² − 9) √𝑥 + 2) /dx]

now expand to d( x²√x -9√x + 2) /dx = d( x².x^1/2 - 9 x^1/2 + 2) / dx = d( x^5/2 -9 x^1/2 + 2) / dx

= (5/2) x^(5/2-1) - (1/2)(9)(x)^1/2-1 + 0 = (5/2) x^3/2 - (1/2)(9)(x)^-1/2 = (5/2) x^3/2 - (4.5) x^-1/2 = (5/2) x √x - 4.5/√x

y' = 2[(x² − 9) √𝑥 + 2] [ 5/2 x √x - 4.5/√x] = [2√𝑥 (x² − 9) + 4] [ 5/2 x √x - 4.5/√x ]

you can further expand this answer by using the foil method, but that won't be necessary because you can leave the answer in this form