Differentiate 1/sinx^2 using chain rule?

2 Answers
May 10, 2017

-2xcos(x^2)/sin^2(x^2)

Explanation:

We can use the chain rule to differentiate the function, the chain rule states d/dx f(g(x)) = f'(g(x)) * g'(x)

So d/dx (sin(x^2))^-1 = -1sin^-2(x^2)*cos(x^2)*2x = -2xcos(x^2)/sin^2(x^2)

May 10, 2017

(df)/(dx)=-2xcotx^2csc^2x^2

Explanation:

We use chain rule here.

Using this in order to differentiate a function of a function, say y, =f(g(x)), where we have to find (dy)/(dx), we need to do (a) substitute u=g(x), which gives us y=f(u). Then we need to use a formula called Chain Rule, which states that (dy)/(dx)=(dy)/(du)xx(du)/(dx).

In fact if we have something like y=f(g(h(x))), we can have (dy)/(dx)=(dy)/(df)xx(df)/(dg)xx(dg)/(dh)

Here we have f(x)=1/(g(x)), where g(x)=sin(h(x)) and h(x)=x^2.

Hence, (df)/(dx)=-1/(sin^2x^2)xxcosx^2xx2x=-2xcotx^2csc^2x^2