Find the 2nd Derivative y = square root of x^2+1

Problem

d2()/(d(x)2)√(,x2+1)

Solution

1. Rewrite the function using a fractional exponent to prepare for differentiation.

y=(x2+1)(1/2)

2. Apply the power rule and chain rule to find the first derivative.

d(y)/d(x)=1/2*(x2+1)(−1/2)⋅2*x

3. Simplify the first derivative by canceling the constants.

d(y)/d(x)=x*(x2+1)(−1/2)

4. Apply the product rule to find the second derivative, where u=x and v=(x2+1)(−1/2)

d2(y)/(d(x)2)=d(x)/d(x)⋅(x2+1)(−1/2)+x⋅d(x2+1)/d(x)

5. Differentiate the components using the chain rule for the second term.

d2(y)/(d(x)2)=1⋅(x2+1)(−1/2)+x⋅(−1/2*(x2+1)(−3/2)⋅2*x)

6. Simplify the expression by combining the x terms and constants.

d2(y)/(d(x)2)=(x2+1)(−1/2)−x2*(x2+1)(−3/2)

7. Factor out the common term (x2+1)(−3/2) to simplify the result into a single fraction.

d2(y)/(d(x)2)=(x2+1)(−3/2)⋅((x2+1)1−x2)

8. Finalize the simplification by subtracting the x2 terms.

d2(y)/(d(x)2)=(x2+1)(−3/2)⋅1

Final Answer

d2(√(,x2+1))/(d(x)2)=1/((x2+1)(3/2))

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