Final answer to the problem
$-1-\cot\left(x\right)^2$
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Step-by-step Solution
Learn how to solve product rule of differentiation problems step by step online. Find the derivative of cos(x)csc(x). Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=\cos\left(x\right) and g=\csc\left(x\right). The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if f(x) = \cos(x), then f'(x) = -\sin(x)\cdot D_x(x). Applying the trigonometric identity: \sin\left(\theta \right)\csc\left(\theta \right) = 1. Taking the derivative of cosecant function: \frac{d}{dx}\left(\csc(x)\right)=-\csc(x)\cdot\cot(x)\cdot D_x(x).
Final answer to the problem
$-1-\cot\left(x\right)^2$
