Find the Derivative - d/dx y=x^2sin(6x)

Problem

d()/d(x)*x2*sin(6*x)

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=x2 and v=sin(6*x)

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

3. Differentiate the first part u=x2 to get d(u)/d(x)=2*x

4. Differentiate the second part v=sin(6*x) using the chain rule to get d(v)/d(x)=6*cos(6*x)

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

(d(x2)*sin(6*x))/d(x)=x2*(6*cos(6*x))+sin(6*x)*(2*x)

6. Simplify the expression by rearranging the terms.

(d(x2)*sin(6*x))/d(x)=6*x2*cos(6*x)+2*x*sin(6*x)

Final Answer

(d(x2)*sin(6*x))/d(x)=6*x2*cos(6*x)+2*x*sin(6*x)

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