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

Problem

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

Solution

1. Identify the rule needed for the expression, which is the product rule for the terms 2*x and cos(x2)

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

3. Set u=2*x and v=cos(x2)

4. Differentiate u to find d(u)/d(x)=2

5. Differentiate v using the chain rule to find d(cos(x2))/d(x)=−sin(x2)⋅2*x=−2*x*sin(x2)

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

2*x*(−2*x*sin(x2))+cos(x2)*(2)

7. Simplify the expression by multiplying the terms.

−4*x2*sin(x2)+2*cos(x2)

Final Answer

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

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