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

Problem

(d(−)*2*x*sin(x2))/d(x)

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=−2*x and v=sin(x2)

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

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

4. Differentiate the second part, v=sin(x2) using the chain rule to get d(v)/d(x)=cos(x2)⋅2*x

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

(d(−)*2*x*sin(x2))/d(x)=−2*x⋅d(sin(x2))/d(x)+sin(x2)⋅(d(−)*2*x)/d(x)

6. Simplify the resulting expression by performing the multiplication.

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

7. Combine terms to reach the final form.

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

Final Answer

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

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