Find the Derivative - d/d@VAR f(x)=6xe^xcsc(x)

Problem

d()/d(x)*6*x*e^x*csc(x)

Solution

1. Identify the function as a product of three terms: 6*x e^x and csc(x)

2. Apply the product rule for three functions, which states d()/d(x)*[u*v*w]=u^′*v*w+u*v^′*w+u*v*w^′

3. Differentiate each component: (d(6)*x)/d(x)=6 d(e^x)/d(x)=e^x and d(csc(x))/d(x)=−csc(x)*cot(x)

4. Substitute these derivatives into the product rule formula:

d()/d(x)*6*x*e^x*csc(x)=(6)*e^x*csc(x)+6*x(e^x)*csc(x)+6*x*e^x*(−csc(x)*cot(x))

5. Factor out the common term 6*e^x*csc(x) to simplify the expression:

d()/d(x)*6*x*e^x*csc(x)=6*e^x*csc(x)*(1+x−x*cot(x))

Final Answer

d()/d(x)*6*x*e^x*csc(x)=6*e^x*csc(x)+6*x*e^x*csc(x)−6*x*e^x*csc(x)*cot(x)

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