Find the 2nd Derivative cos(x^2)

Problem

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

Solution

1. Identify the function ƒ(x)=cos(x2) and recognize that finding the second derivative requires applying the chain rule twice.

2. Apply the chain rule to find the first derivative, where the outer function is cos(u) and the inner function is u=x2

d(cos(x2))/d(x)=−sin(x2)⋅2*x

3. Simplify the first derivative expression.

d(cos(x2))/d(x)=−2*x*sin(x2)

4. Apply the product rule to find the second derivative, using the formula d()/d(x)*[u*v]=ud(v)/d(x)+vd(u)/d(x) with u=−2*x and v=sin(x2)

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

5. Differentiate the individual components using the chain rule for sin(x2) and the power rule for −2*x

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

6. Simplify the resulting expression by multiplying the terms.

d2(cos(x2))/(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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