Given: #f(x) = ((2x^2 + x)^3(x))/sqrt(x)#
Simplify to get rid of the denominator:
#f(x) = (2x^2 + x)^3 *x*x^(-1/2) = (2x^2 + x)^3 x^(1/2)#
Use the power rule/chain rule: #(u^n)' = n(u^(n-1))u'#
Let #u = 2x^2 + x; " "n = 3; " "(u^3)' = 3(2x^2 + x)^2 (4x + 1)#
Since #f(x) = (2x^2 + x)^3 x^(1/2)#,
Use the product rule: #(vw)' = v w' + w v'#
#v = (2x^2 + x)^3; " "v' = 3(2x^2 + x)^2 (4x + 1)#
#w = sqrt(x) = x^(1/2); " "w' = 1/2 x^(-1/2) = 1/(2sqrt(x))#
#f'(x) = (2x^2 + x)^3*1/(2sqrt(x)) + sqrt(x) *3(2x^2 + x)^2 (4x + 1)#
Factor the GCF (greatest common factor):
#f'(x) = (2x^2 + x)^2 [(2x^2 + x)/(2sqrt(x)) + 3sqrt(x)(4x + 1)]#
Simplify:
#f'(x) = [x(2x+1)]^2[(2x^2 + x)/(2sqrt(x)) + (3sqrt(x)* 2sqrt(x))/(2 sqrt(x)) *(4x + 1)]#
#f'(x) = x^2(2x+1)^2[(2x^2 + x)/(2sqrt(x)) + (6x)/(2 sqrt(x)) *(4x + 1)]#
#f'(x) = x^2(2x+1)^2 1/(2sqrt(x))[(2x^2 + x) + 6x(4x + 1)]#
#f'(x) = 1/2 x^(3/2)(2x+1)^2[2x^2 + x + 24x^2 + 6x]#
#f'(x) = 1/2 x^(3/2)(2x+1)^2 (26x^2 +7x)#
#f'(x) = 1/2 x^(3/2)(2x+1)^2 x(26x +7)#
#f'(x) = 1/2 x^(5/2)(2x+1)^2 (26x +7)#
