Find the Derivative - d/dx xsin(3x)

Problem

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

Solution

1. Identify the rule needed for the expression x*sin(3*x) which is the product of two functions u=x and v=sin(3*x)

2. Apply the product rule, which states d()/d(x)*u*v=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=sin(3*x) using the chain rule, which gives d(sin(3*x))/d(x)=cos(3*x)⋅3=3*cos(3*x)

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

x*(3*cos(3*x))+sin(3*x)*(1)

6. Simplify the resulting expression.

3*x*cos(3*x)+sin(3*x)

Final Answer

(d(x)*sin(3*x))/d(x)=3*x*cos(3*x)+sin(3*x)

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