How do you find the derivative of #y = log_3 [(x+1/x-1)^ln3]#?

1 Answer
Bdub
Mar 31, 2017

see below

Explanation:

Use the Property #color(red)(log_b x^n =n log_bx# to simplify the problem first and then use the the formula #color(red)(d/dx(log_b f(x))=1/(f(x)ln b) * f'(x)# to find the derivative

#y=log_3[(x+1/x-1)^ln3]#

#y=ln3* log_3(x+1/x-1) #

#color(blue)(y'=ln 3* 1/((x+1/x -1)ln 3)*(1-1/x^2)#

#color(blue)(y'=cancel ln 3* 1/((x+1/x -1)cancel ln 3)*(1-1/x^2)#

#color(blue)(y'= 1/((x+1/x -1)) *(1-1/x^2)#

#color(blue)(y'= 1/(((x^2+1-x)/x )) *((x^2-1)/x^2)#

#color(blue)(y'= x/(x^2-x+1) *(x^2-1)/(x^2)#

#color(blue)(y'= cancelx/(x^2-x+1) *(x^2-1)/(x^cancel2)#

#color(blue)(y'= (x^2-1)/(x^3-x^2+x) #

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