Find the Derivative - d/dx xcsc(x)

Problem

d()/d(x)*x*csc(x)

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=x and v=csc(x)

2. Apply the product rule, which states that d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the individual components: d(x)/d(x)=1 and d(csc(x))/d(x)=−csc(x)*cot(x)

4. Substitute these derivatives back into the product rule formula.

d()/d(x)*x*csc(x)=x*(−csc(x)*cot(x))+csc(x)*(1)

5. Simplify the expression by factoring out the common term csc(x)

d()/d(x)*x*csc(x)=csc(x)−x*csc(x)*cot(x)

d()/d(x)*x*csc(x)=csc(x)*(1−x*cot(x))

Final Answer

d()/d(x)*x*csc(x)=csc(x)*(1−x*cot(x))

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