How do you find the first and second derivative of #(x^3-3x^2+8x+18)/x#? Calculus Basic Differentiation Rules Quotient Rule 2 Answers Rhys Jun 12, 2018 Shown below Explanation: #y=(x^3-3x^2 + 8x+18 )/x # # y= x^3/x - (3x^2)/x + (8x)/x + 18/x # #y = x^2 -3x + 8 +18x^(-1) # #=> d/dx((x^3-3x^2 + 8x+18 )/x) = d/dx (x^2 -3x + 8 +18x^(-1) )# Use the power rule: #color(red)(d/dx ( x^n ) = nx^(n-1) # #=> (dy)/(dx) = 2x - 3 -18x^(-2) # #=> (d^2y)/(dx^2) = d/dx ( (dy)/(dx) ) # #=> (d^2y)/(dx^2) = d/dx ( 2x - 3 -18x^(-2) )# #=> (d^2y)/(dx^2) = 2 + 36x^(-3) # Answer link Ratnaker Mehta Jun 12, 2018 # (i) : f'(x)=2x-3-18/x^2=(2x^3-3x-18)/x^2#. # (ii) : f''(x)=2+36/x^3={2(x^3+18)}/x^3#. Explanation: Prerequisite : #(x^n)'=nx^(n-1)#. Let #f(x)=(x^3-3x^2+8x+18)/x#. #:. f(x)=x^3/x-(3x^2)/x+(8x)/x+18/x, i.e., # # f(x)=x^2-3x+8+18/x#. #:. f'(x)=(x^2)'-3(x)'+0+18(x^-1)'#, #=2x^(2-1)-3(1x^(1-1))+18(-1x^(-1-1))#. # rArr f'(x)=2x-3-18/x^2=(2x^3-3x-18)/x^2#. Similarly, #f''(x)=(f'(x))'#, #=(2x-3-18x^-2)'#, #=2-0-18(-2x^(-2-1))#. # rArr f''(x)=2+36/x^3={2(x^3+18)}/x^3#. Answer link Related questions What is the Quotient Rule for derivatives? How do I use the quotient rule to find the derivative? How do you prove the quotient rule? How do you use the quotient rule to differentiate #y=(2x^4-3x)/(4x-1)#? How do you use the quotient rule to differentiate #y=cos(x)/ln(x)#? How do you use the quotient rule to find the derivative of #y=tan(x)# ? How do you use the quotient rule to find the derivative of #y=x/(x^2+1)# ? How do you use the quotient rule to find the derivative of #y=(e^x+1)/(e^x-1)# ? How do you use the quotient rule to find the derivative of #y=(x-sqrt(x))/(x^(1/3))# ? How do you use the quotient rule to find the derivative of #y=x/(3+e^x)# ? See all questions in Quotient Rule Impact of this question 1732 views around the world You can reuse this answer Creative Commons License