Question

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

Answer 1

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})`