Find the Derivative - d/dx xcos(2x)

Problem

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

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=x and v=cos(2*x)

2. Apply the product rule, which states (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the first part u=x to get d(x)/d(x)=1

4. Differentiate the second part v=cos(2*x) using the chain rule, which gives d(cos(2*x))/d(x)=−sin(2*x)⋅2=−2*sin(2*x)

5. Substitute these derivatives back into the product rule formula.

x*(−2*sin(2*x))+cos(2*x)*(1)

6. Simplify the resulting expression by rearranging the terms.

−2*x*sin(2*x)+cos(2*x)

Final Answer

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

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