Find the Derivative - d/dx y=xcos(3x)

Problem

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

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x and cos(3*x) use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the variables for the product rule. Let u=x and v=cos(3*x)

3. Differentiate each part. The derivative of u is d(x)/d(x)=1 To find the derivative of v apply the chain rule: d(cos(3*x))/d(x)=−sin(3*x)⋅(d(3)*x)/d(x)=−3*sin(3*x)

4. Apply the product rule formula by substituting the parts back in.

d(y)/d(x)=x*(−3*sin(3*x))+cos(3*x)*(1)

5. Simplify the resulting expression by rearranging the terms.

d(y)/d(x)=−3*x*sin(3*x)+cos(3*x)

Final Answer

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

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