# How to adapt and use the leibniz notation chain rule to answer the following question? Show all steps to properly solve this question.given y=(4u-1)^3, u=9v^-2+v^-1/2, v= 3w/w-2 and w= -2sqrtx +1....

How to adapt and use the leibniz notation chain rule to answer the following question? Show all steps to properly solve this question.

given y=(4u-1)^3, u=9v^-2+v^-1/2, v= 3w/w-2 and w= -2sqrtx +1. A general formula for dy/dx in leibniz notation, adapting the chain rule for use with this triple composition can be written as dy/dx=dy/du*du/dv*dv/dw*dw/dx. evaluate dy/dx if x=1. Answer in exact value

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### 1 Answer

You need to use chain rule to find` (dy)/(dx)` since the function y yields as a result of composition of functions u,v,w.

`(dy)/(dx)=(dy)/(du)*(du)/(dv)*(dv)/(dw)*(dw)/(dx)` `(dy)/(dx) = (d((4u-1)^3))/(du)* (d(9v^-2+v^(-1/2)))/(dv)* (d(3w/(w-2)))/(dw)* (d(-2sqrt(x) +1))/(dx)`

`` `(dy)/(dx) = 12(4u-1)^2*(-18v^(-3) - (1/2)v^(-3/2))*((3(w-2)-3w)/(w-2)^2)*(-1/sqrtx)`

You need to evaluate`(dw)/(dx)` at x=1 such that:

`(dw)/(dx)|_(x=1) = -1/sqrt1 = -1`

You need to evaluate the value of function w at x=1 such that:

`w=-2+1=-1`

`` `(dv)/(dw) = (3(w-2)-3w)/(w-2)^2 =gt (dv)/(dw) = -6/(w-2)^2`

You need to substitute -1 for w such that:

`(dv)/(dw) = -6/(-1-2)^2 = -6/9 = -2/3`

You need to evaluate the value of function `v= 3w/(w-2)` at w=-1 such that:

`v= -3/(-1-2)=1`

`(du)/(dv) = -18 - (1/2) = -37/2`

`` You need to evaluate the value of function , `u=9v^-2+v^(-1/2)` at v=1 such that:

`u = 9+1 = 10` `(dy)/(du)|_(u=10) = 12(4*10-1)^2 = 18252`

**Hence, substituting the values above in `(dy)/(dx) = (dy)/(du)* (du)/(dv)* (dv)/(dw)* (dw)/(dx)` yields: `(dy)/(dx) = 18252* (-37/2)* (-2/3)* (-1)=-36.` **