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

Problem

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

Solution

1. Identify the outer function and the inner function to apply the chain rule. The expression is of the form u2 where u=sin(2*x)

2. Apply the power rule to the outer function u2 which gives 2*u⋅d(u)/d(x)

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

3. Apply the chain rule again to differentiate sin(2*x) The derivative of sin(v) is cos(v)⋅d(v)/d(x) where v=2*x

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

4. Differentiate the innermost function 2*x with respect to x which equals 2

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

5. Substitute the results back into the expression.

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

6. Simplify the expression by multiplying the constants.

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

7. Apply the double angle identity sin(2*θ)=2*sin(θ)*cos(θ) if further simplification is desired, where θ=2*x

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

Final Answer

d(sin(2*x))/d(x)=2*sin(4*x)

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