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

Problem

d(cos(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=cos(2*x)

2. Apply the Power Rule to the outer function. The derivative of u2 is 2*u⋅d(u)/d(x)

3. Differentiate the inner function cos(2*x) This requires a second application of the Chain Rule because the argument of the cosine is 2*x

4. Apply the derivative of cosine, which is −sin(u) and multiply by the derivative of the innermost argument 2*x which is 2

5. Combine all the components together.

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

6. Simplify the expression by multiplying the constants.

d(cos(2*x))/d(x)=−4*cos(2*x)*sin(2*x)

7. Apply the double angle identity 2*sin(θ)*cos(θ)=sin(2*θ) to further simplify the result.

−2*(2*sin(2*x)*cos(2*x))=−2*sin(4*x)

Final Answer

d(cos(2*x))/d(x)=−2*sin(4*x)

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