Find dy/dx y=x^2sin(8x)

Problem

d()/d(x)*x2*sin(8*x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x2 and sin(8*x) use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions for the product rule. Let u=x2 and v=sin(8*x)

3. Differentiate each part. The derivative of u is d(x2)/d(x)=2*x The derivative of v requires the chain rule: d(sin(8*x))/d(x)=cos(8*x)⋅(d(8)*x)/d(x)=8*cos(8*x)

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

d(y)/d(x)=x2⋅8*cos(8*x)+sin(8*x)⋅2*x

5. Simplify the expression by rearranging the terms and factoring out common factors if possible.

d(y)/d(x)=8*x2*cos(8*x)+2*x*sin(8*x)

Final Answer

d(y)/d(x)=8*x2*cos(8*x)+2*x*sin(8*x)

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