Find the Derivative - d/dx y=x^2sin(4x)

Problem

d()/d(x)*x2*sin(4*x)

Solution

1. Identify the rule needed for the expression, which is the product of two functions: u=x2 and v=sin(4*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 using the power rule.

d(x2)/d(x)=2*x

4. Differentiate the second part, v=sin(4*x) using the chain rule.

d(sin(4*x))/d(x)=cos(4*x)⋅(d(4)*x)/d(x)

d(sin(4*x))/d(x)=4*cos(4*x)

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

d(y)/d(x)=x2*(4*cos(4*x))+sin(4*x)*(2*x)

6. Simplify the expression by rearranging the terms.

d(y)/d(x)=4*x2*cos(4*x)+2*x*sin(4*x)

Final Answer

d()/d(x)*x2*sin(4*x)=4*x2*cos(4*x)+2*x*sin(4*x)

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