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if the function is in the form y=f(u) and u=g(x), then find the dy/dx as a function of x. `y=(2x+1)^5`

Hello!


In general, if `y(x)=f(u)` and `u=g(x),` then `(dy)/(dx)=(df)/(du)*(dg)/(dx),` or `y'(x)=f'(g(x))*g'(x).`


In the given case, for `y(x)=(2x+1)^5,` we can use `f(u)=u^5` and `g(x)=2x+1,` then as required `y(x)=f(g(x)).`


It is simple to compute the required derivatives, they are both from the table of well-known ones: `f'(u)=5u^4,` `g'(x)=2.` Therefore


`(dy)/(dx)=5u^4*2=5(2x+1)^4*2=10(2x+1)^4.`


This is the answer.

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