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

Problem

d()/d(x)*(−csc(x)−sin(x))

Solution

1. Apply the sum rule for differentiation, which allows the derivative of a sum or difference to be taken term by term.

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

2. Apply the constant multiple rule to pull the negative sign out of the first term.

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

3. Differentiate the cosecant function using the standard trigonometric derivative formula d(csc(x))/d(x)=−csc(x)*cot(x)

−d(csc(x))/d(x)=−(−csc(x)*cot(x))

4. Differentiate the sine function using the standard trigonometric derivative formula d(sin(x))/d(x)=cos(x)

d(sin(x))/d(x)=cos(x)

5. Combine and simplify the results from the previous steps.

csc(x)*cot(x)−cos(x)

Final Answer

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

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