Find the Derivative - d/dx 1+cos(x)^2

Problem

d()/d(x)*(1+cos2(x))

Solution

1. Apply the sum rule for differentiation, which allows us to differentiate each term of the expression separately.

d()/d(x)*(1+cos2(x))=d(1)/d(x)+d(cos2(x))/d(x)

2. Differentiate the constant term. The derivative of any constant with respect to x is 0

d(1)/d(x)=0

3. Apply the power rule and the chain rule to the second term. For a function in the form un the derivative is n*u(n−1)⋅d(u)/d(x)

d(cos2(x))/d(x)=2*cos(x)⋅d(cos(x))/d(x)

4. Differentiate the trigonometric function cos(x) The derivative of cos(x) is −sin(x)

d(cos(x))/d(x)=−sin(x)

5. Substitute and simplify the expression by combining the results from the previous steps.

0+2*cos(x)*(−sin(x))=−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.

−2*sin(x)*cos(x)=−sin(2*x)

Final Answer

d()/d(x)*(1+cos2(x))=−sin(2*x)

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