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

Problem

(d(2)*sin2(x))/d(x)

Solution

1. Identify the constant multiple rule to move the coefficient 2 outside the derivative.

(d(2)*sin2(x))/d(x)=2d(sin2(x))/d(x)

2. Apply the power rule and the chain rule to the expression sin2(x) treating it as (sin(x))2

d(sin2(x))/d(x)=2*sin(x)⋅d(sin(x))/d(x)

3. Differentiate the inner function sin(x) to get cos(x)

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

4. Substitute the result back into the expression.

2⋅(2*sin(x)*cos(x))=4*sin(x)*cos(x)

5. Simplify using the double angle identity sin(2*x)=2*sin(x)*cos(x) if desired.

4*sin(x)*cos(x)=2*sin(2*x)

Final Answer

(d(2)*sin2(x))/d(x)=2*sin(2*x)

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