Find the Derivative - d/d@VAR f(x)=cos(x)^2

Problem

d(cos2(x))/d(x)

Solution

1. Identify the function as a composition of functions where the outer function is the power u2 and the inner function is u=cos(x)

2. Apply the power rule to the outer function, which states that the derivative of un is n*u(n−1)⋅d(u)/d(x)

3. Differentiate the inner function cos(x) which results in −sin(x)

4. Combine the results using the chain rule.

d(cos2(x))/d(x)=2*cos(x)⋅(−sin(x))

5. Simplify the expression by moving the negative sign to the front.

−2*sin(x)*cos(x)

6. Apply the double angle identity sin(2*x)=2*sin(x)*cos(x) to write the result in a more compact form.

Final Answer

d(cos2(x))/d(x)=−sin(2*x)

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