How do you differentiate #f(x)= (x^3-1)/( x^2+3x) ^ (3/2) # using the quotient rule?

1 Answer
Binayaka C.
Oct 8, 2017

# f^'(x) = (3x^2(x^2+3x)-3/2(x^3-1)(2x+3)) /((x^2+3x)^(5/2))#

Explanation:

#f(x) = (x^3-1)/(x^2+3x)^(3/2)# .Quotient rule: #d/dx(f/g)=(gf^'-fg^')/g^2#

# f^'(x) = (3x^2(x^2+3x)^(3/2)-(x^3-1)*3/2(x^2+3x)^(3/2-1)(2x+3)) /(((x^2+3x)^(3/2))^2#

# f^'(x) = (3x^2(x^2+3x)^(3/2)-(x^3-1)*3/2(x^2+3x)^(1/2)(2x+3)) /((x^2+3x)^3)#

# f^'(x) = ((x^2+3x)^(1/2)(3x^2(x^2+3x)-(x^3-1)*3/2(2x+3))) /((x^2+3x)^3)#

# f^'(x) = (3x^2(x^2+3x)-3/2(x^3-1)(2x+3)) /((x^2+3x)^(5/2))# [Ans]

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