# If #f(x)= csc 7 x # and #g(x) = 3x^2 -5 #, how do you differentiate #f(g(x)) # using the chain rule?

##### 1 Answer

#### Answer:

#### Explanation:

First, note the chain rule states that

#d/dx[f(g(x))]=f'(g(x))*g'(x)#

Let's focus on the first part,

We must find

In the case of a cosecant function, the chain rule states that

#d/dx(csc(h(x)))=-csc(h(x))cot(h(x))*h'(x)#

Thus, since in

#f'(x)=-7csc(7x)cot(7x)#

Thus to find

#f'(g(x))=-7csc(21x^2-35)cot(21x^2-35)#

Now, we should find the second term of the original chain rule expression,

#g'(x)=6x#

Multiplying

#d/dx(f(g(x)))=-42xcsc(21x^2-35)cot(21x^2-35)#