Find the Derivative - d/dx f(x)=2cos(x)+sin(2x)

Problem

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

Solution

1. Apply the sum rule for derivatives to differentiate each term separately.

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

2. Apply the constant multiple rule to the first term, noting that the derivative of cos(x) is −sin(x)

(d(2)*cos(x))/d(x)=−2*sin(x)

3. Apply the chain rule to the second term, where the outer function is sin(u) and the inner function is u=2*x

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

4. Differentiate the inner function and multiply to complete the chain rule step.

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

5. Combine the results of the individual derivatives into a single expression.

−2*sin(x)+2*cos(2*x)

Final Answer

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

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