Find the 2nd Derivative 2sin(x)+sin(2x)

Problem

d2()/(d(x)2)*(2*sin(x)+sin(2*x))

Solution

1. Identify the function to be differentiated twice, which is ƒ(x)=2*sin(x)+sin(2*x)

2. Find the first derivative by applying the sum rule and the chain rule to the second term.

ƒ(x)′=(d(2)*sin(x))/d(x)+d(sin(2*x))/d(x)

3. Apply differentiation rules where the derivative of sin(x) is cos(x) and the derivative of sin(2*x) is 2*cos(2*x)

ƒ(x)′=2*cos(x)+2*cos(2*x)

4. Find the second derivative by differentiating ƒ(x)′ with respect to x

ƒ(x)″=(d(2)*cos(x))/d(x)+(d(2)*cos(2*x))/d(x)

5. Apply differentiation rules again where the derivative of cos(x) is −sin(x) and the derivative of 2*cos(2*x) is −4*sin(2*x)

ƒ(x)″=−2*sin(x)−4*sin(2*x)

Final Answer

d2()/(d(x)2)*(2*sin(x)+sin(2*x))=−2*sin(x)−4*sin(2*x)

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