# How do you differentiate #f(x)=csc(1/x^3) # using the chain rule?

##### 1 Answer

#### Answer:

# f'(x) = (3csc(1/x^3)cot(1/x^3))/x^4 #

#### Explanation:

f you are studying maths, then you should learn the Chain Rule for Differentiation, and practice how to use it:

If

# y=f(x) # then# f'(x)=dy/dx=dy/(du)(du)/dx #

I was taught to remember that the differential can be treated like a fraction and that the "

# dy/dx = dy/(dv)(dv)/(du)(du)/dx # etc, or# (dy/dx = dy/color(red)cancel(dv)color(red)cancel(dv)/color(blue)cancel(du)color(blue)cancel(du)/dx) #

So with

# { ("Let "u=1/x^3, => , (du)/dx=-3/x^4), ("Then "y=cscu, =>, dy/(du)=-cscucotu ) :}#

Using

# \ \ \ \ \ dy/dx = (-cscucotu)(-3/x^4) #

# \ \ \ \ \ \ \ \ \ \ = (3cscucotu)/x^4 #

# \ \ \ \ \ \ \ \ \ \ = (3csc(1/x^3)cot(1/x^3))/x^4 #