Find the 2nd Derivative y=cos(X^2)

Problem

d2()/(d(x)2)*cos(x2)

Solution

1. Identify the function as y=cos(x2) and note that finding the second derivative requires applying the chain rule twice.

2. Apply the chain rule to find the first derivative d(y)/d(x) The outer function is cos(u) and the inner function is u=x2

3. Differentiate the outer function to get −sin(x2) and multiply by the derivative of the inner function, which is 2*x

d(y)/d(x)=−sin(x2)⋅2*x

d(y)/d(x)=−2*x*sin(x2)

4. Apply the product rule to find the second derivative d2(y)/(d(x)2) Let u=−2*x and v=sin(x2) The formula is d(u*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

5. Calculate the derivatives of the parts: d(u)/d(x)=−2 and d(v)/d(x)=cos(x2)⋅2*x (using the chain rule again).

6. Substitute these into the product rule formula.

d2(y)/(d(x)2)=(−2*x)*(2*x*cos(x2))+(sin(x2))*(−2)

7. Simplify the expression by multiplying the terms.

d2(y)/(d(x)2)=−4*x2*cos(x2)−2*sin(x2)

Final Answer

d2(cos(x2))/(d(x)2)=−4*x2*cos(x2)−2*sin(x2)

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