Find the Derivative - d/dx thetacos(theta)sin(theta)

Problem

(d(θ)*cos(θ)*sin(θ))/d(θ)

Solution

1. Apply the double angle identity for sine, sin(2*θ)=2*sin(θ)*cos(θ) to rewrite the expression as 1/2*θ*sin(2*θ)

2. Apply the product rule, (d(u)*v)/d(θ)=ud(v)/d(θ)+vd(u)/d(θ) where u=1/2*θ and v=sin(2*θ)

3. Differentiate the first part of the product rule using the chain rule for sin(2*θ) which results in 2*cos(2*θ)

4. Differentiate the second part of the product rule, where the derivative of 1/2*θ is 1/2

5. Combine the terms to get 1/2*θ*(2*cos(2*θ))+1/2*sin(2*θ)

6. Simplify the expression by distributing the constants.

Final Answer

(d(θ)*cos(θ)*sin(θ))/d(θ)=θ*cos(2*θ)+1/2*sin(2*θ)

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