Find the Derivative - d/dx xsin(x^2)

Problem

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

Solution

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

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

3. Set u=x and v=sin(x2)

4. Differentiate u to get d(x)/d(x)=1

5. Differentiate v using the chain rule, where the derivative of sin(u) is cos(u)⋅d(u)/d(x)

6. Calculate d(sin(x2))/d(x)=cos(x2)⋅2*x=2*x*cos(x2)

7. Combine the parts using the product rule formula.

x⋅2*x*cos(x2)+sin(x2)⋅1

8. Simplify the expression by multiplying the terms.

2*x2*cos(x2)+sin(x2)

Final Answer

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

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