How do you differentiate #f(x)=sqrtsin((3x+7)^3)# using the chain rule.?

1 Answer
Guilherme N.
Dec 21, 2015

Using chain rule consists in dealing with the separate operations in the equation one at a time.

Explanation:

First, chain rule states that #(dy)/(dx)=(dy)/(du)(du)/(dx)#, or, in this case, we'll use a longer one, following the same logic: #(dy)/(dx)=(dy)/(du)(du)/(dv)(dv)/(dw)(dw)/(dx)#

Here, we can rename #y=sqrt(u)#, #u=sin(v)#, #v=(w)^3# and #w=3x+7#

Following the statement of the chain rule:

#(dw)/(dx)=3#

#(dv)/(dw)=3w^2#

#(du)/(dv)=cos(v)#

#(dy)/(du)=1/(2sqrt(u))#

Aggregating them...

#(dy)/(dx)=1/(2sqrt(u))(cos(v))(3w^2)(3)#

Substituting #u#, #v# and #w#:

#(dy)/(dx)=(cos(w^3)(3(3x+7)))/(2sqrt(sin(v)))#

#(dy)/(dx)=(cos(3x+7)^2(9x+21))/(2sqrt(sin(w^3)))#

#(dy)/(dx)=(cos(3x+7)^2(9x+21))/(2sqrt(sin(3x+7)^3))#

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