Find the Derivative - d/dx csc(x)cot(x)

Problem

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

Solution

1. Identify the rule needed for the derivative of a product of two functions, which is the product rule.

(d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions to the variables u and v

u=csc(x)

v=cot(x)

3. Differentiate each component individually using standard trigonometric derivative rules.

d(csc(x))/d(x)=−csc(x)*cot(x)

d(cot(x))/d(x)=−csc2(x)

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

(d(csc(x))*cot(x))/d(x)=csc(x)*(−csc2(x))+cot(x)*(−csc(x)*cot(x))

5. Simplify the expression by multiplying the terms and factoring out common factors.

(d(csc(x))*cot(x))/d(x)=−csc3(x)−csc(x)*cot2(x)

(d(csc(x))*cot(x))/d(x)=−csc(x)*(csc2(x)+cot2(x))

Final Answer

(d(csc(x))*cot(x))/d(x)=−csc3(x)−csc(x)*cot2(x)

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