How do you differentiate #f(x)=csc(e^(x^2-5x)) # using the chain rule?

1 Answer
Ananda Dasgupta
Jul 14, 2016

You differentiate the outer function, multiply by the derivative of the inner one and go on ...

Explanation:

The derivative of #csc(u)# with respect to #u# is #-cot(u)csc(u)#. In this problem #u=e^{x^2-5x}#, Since we want to differentiate with respect to #x#, rather than with respect to #u#, we must multiply the result by #{du}/dx# (this, after all, is the chain rule). We have

#{du}/{dx} = (2 x-5) e^{x^2-5x}#
(where we have applied the chain rule again)
Putting all this together, we get

# d/{dx} (csc(e^{x^2-5x})) = - (2 x-5) e^{x^2-5x} csc( e^{x^2-5x})cot( e^{x^2-5x})#

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