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

Question

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

Ezra Adams · Accepted Answer

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

First, note the chain rule states that

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

Let's focus on the first part, #f'(g(x))#.

We must find #f'(x)#. Ironically, in order to do so, the chain rule must be used once more.

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 #csc(7x)# we see that #h(x)=7x#, and #h'(x)=7#,

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

Thus to find #f'(g(x))# we plug #g(x)# into every #x# in #f'(x)#:

#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)#. This requires only the power rule.

#g'(x)=6x#

Multiplying #f'(g(x))# and #g'(x)# we see that the derivative of the entire composite function is

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

Samantha Adkison · Answer

undefined To differentiate $ f(g(x)) $ using the chain rule, follow these steps:

1. Find $ f'(x) $ and $ g'(x) $.
2. Substitute $ g(x) $ into $ f(x) $ to get $ f(g(x)) $.
3. Differentiate $ f(g(x)) $ with respect to $ x $ using the chain rule, which states that $ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) $.

Now, let's apply these steps:

1. Find $ f'(x) $ and $ g'(x) $:
   - $ f'(x) = -\csc(x) \cot(x) $
   - $ g'(x) = 6x $

2. Substitute $ g(x) $ into $ f(x) $:
   - $ f(g(x)) = \csc(3x^2 - 5) $

3. Apply the chain rule:
   - $ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) $
   - $ = -\csc(3x^2 - 5) \cot(3x^2 - 5) \cdot 6x $

So, the derivative of $ f(g(x)) $ with respect to $ x $ is $ -6x\csc(3x^2 - 5) \cot(3x^2 - 5) $.