Find the derivative of the function. f(x) =

Question

f(x) =(x³ − 5x² − 4x )/ x^1/2

Answer 1

f(x) =(x³ − 5x² − 4x )/ x^1/2

Calculate the derivative of the function using the quotient rule.

d/dx(u/v) =[v(du/dx) −u(dv/dx)]/v^2

f '(x) =[x^1/2*d/dx(x³ − 5x² − 4x ) − (x³ − 5x² − 4x )*d/dx(x^1/2 )] /(x^1/2 )^2

         ⁼[x^1/2*(3x² − 10x − 4) − (x³ − 5x² − 4x)*(1/2)(x^−1/2 )] /(x)

        = [(3x^5/2 − 10x^3/2 − 4x^1/2 ) − (x^5/2 − 5x^3/2 − 4x^1/2 )*(1/2)] /(x)

     = [(3x^5/2 − 10x^3/2 − 4x^1/2 ) − (x^5/2 − 5x^3/2 − 4x^1/2 )*(1/2)] /(x)

     = [(5/2)(x^5/2) − (15/2)(x^3/2)− (2x^1/2)]/x

     =[(5/2)(x^3/2) − (15/2)(x^1/2)− (2x^-1/2)]