How do you differentiate #f(x)= (2 x^2 + 7 x - 6 )/ (x sinx )# using the quotient rule?

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Answer 1 · 2018-01-24T12:27:25

# f'(x)=(2sinx(x^2+3)-xcosx(2x^2+7x-6))/((xsinx)^2)#

#f(x)=(2x^2+7x-6)/(xsinx)#

Quotient rule: #d/dx(f/g)=(gf^'-fg^')/g^2#

#d/dx(2x^2+7x-6)= 4x+7#

#d/dx(xsinx)= sinx+xcosx#

#f'(x)=(xsinx(4x+7)-(2x^2+7x-6)(sinx+xcosx))/(xsinx)^2#

#=(4x^2sinx+cancel(7xsinx)-2x^2sinx-cancel(7xsinx)+6sinx-2x^3cosx-7x^2cosx+6xcosx)/((xsinx)^2)#

#=(2x^2sinx+6sinx-xcosx(2x^2+7x-6))/((xsinx)^2)#

#=(2sinx(x^2+3)-xcosx(2x^2+7x-6))/((xsinx)^2)#

#:. f'(x)=(2sinx(x^2+3)-xcosx(2x^2+7x-6))/((xsinx)^2)# [Ans]