If #f(x)= sec 9x # and #g(x) = sqrt(2x-3 #, how do you differentiate #f(g(x)) # using the chain rule?

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

1. Compute the derivative of ( f(x) ) with respect to ( x ).
2. Compute the derivative of ( g(x) ) with respect to ( x ).
3. Substitute ( g(x) ) into ( f(x) ).
4. Multiply the derivatives obtained in steps 1 and 2.
5. Simplify the expression obtained in step 4.

Given ( f(x) = \sec(9x) ) and ( g(x) = \sqrt{2x - 3} ):

1. The derivative of ( f(x) = \sec(9x) ) with respect to ( x ) is ( f'(x) = 9\sec(9x)\tan(9x) ).
2. The derivative of ( g(x) = \sqrt{2x - 3} ) with respect to ( x ) is ( g'(x) = \frac{1}{2\sqrt{2x - 3}} ).
3. Substitute ( g(x) ) into ( f(x) ): ( f(g(x)) = \sec(9\sqrt{2x - 3}) ).
4. Multiply the derivatives: ( f'(g(x)) \cdot g'(x) = 9\sec(9\sqrt{2x - 3})\tan(9\sqrt{2x - 3}) \cdot \frac{1}{2\sqrt{2x - 3}} ).
5. Simplify the expression obtained in step 4.