Find the Derivative - d/dx xsin(x)^2

Problem

d()/d(x)*x*sin2(x)

Solution

1. Identify the rule needed for the product of two functions, x and sin2(x) which is the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Apply the product rule by setting u=x and v=sin2(x)

3. Differentiate the first part, d(x)/d(x)=1

4. Differentiate the second part, sin2(x) using the chain rule: d(sin2(x))/d(x)=2*sin(x)*cos(x)

5. Combine the results into the product rule formula.

(d(x)*sin2(x))/d(x)=xd(sin2(x))/d(x)+sin2(x)d(x)/d(x)

(d(x)*sin2(x))/d(x)=x*(2*sin(x)*cos(x))+sin2(x)*(1)

6. Simplify the expression using the double angle identity 2*sin(x)*cos(x)=sin(2*x)

(d(x)*sin2(x))/d(x)=x*sin(2*x)+sin2(x)

Final Answer

d()/d(x)*x*sin2(x)=x*sin(2*x)+sin2(x)

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