Find the Derivative - d/dx y=cos(2x)sin(3x)

Problem

d()/d(x)*cos(2*x)*sin(3*x)

Solution

1. Identify the product rule for differentiation, which states that for two functions u and v the derivative is ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions u=cos(2*x) and v=sin(3*x)

3. Differentiate u using the chain rule, where d(cos(2*x))/d(x)=−2*sin(2*x)

4. Differentiate v using the chain rule, where d(sin(3*x))/d(x)=3*cos(3*x)

5. Apply the product rule formula by substituting the functions and their derivatives.

y′=cos(2*x)⋅3*cos(3*x)+sin(3*x)⋅(−2*sin(2*x))

6. Simplify the expression by rearranging the constants and terms.

y′=3*cos(2*x)*cos(3*x)−2*sin(2*x)*sin(3*x)

Final Answer

d()/d(x)*cos(2*x)*sin(3*x)=3*cos(2*x)*cos(3*x)−2*sin(2*x)*sin(3*x)

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