How do you use chain rule to find the derivative of $y = 2 csc^3(sqrt(x))$ ?

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Answer 1 · 2015-03-01T01:13:17

You can easily illustrate the chain rule using Leibniz notation

Let $y=2u^3$ then $dy/(du)=6u^2$

Let $u=csc(w)$ then $(du)/(dw)=-csc(w)cot(w)$

Let $w=sqrt(x)$ then $(dw)/(dx)=1/(2sqrt(x))$

Now the chain rule is

$dy/dx=(dy)/(du)(du)/(dw)(dw)/(dx)$

$dy/dx=6u^2(-csc(w)cot(w))1/(2sqrt(x))$

Remember that $u=csc(w)$ and $w=sqrt(x)$

$dy/dx=6csc^2(sqrt(x))(-csc(sqrt(x))cot(sqrt(x)))(1/(2sqrt(x)))$

Some simplifying

$dy/dx=(-3csc^3(sqrt(x))cot(sqrt(x)))/sqrt(x)$

How do you use chain rule to find the derivative of y = 2 csc^3(sqrt(x)) y = 2 csc^3(sqrt(x)) ?