Differentiate #1/sinx^2# using chain rule?
↳Redirected from
"What polyatomic ion has a positive charge?"
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)#
#(df)/(dx)=-2xcotx^2csc^2x^2#
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#
