If y = √(x^2 + 1), find dy/dx at x = 1.

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Question: If y = √(x^2 + 1), find dy/dx at x = 1.

Options:

Correct Answer: 1/√2

Solution:

dy/dx = (1/2)(x^2 + 1)^(-1/2)(2x). At x = 1, dy/dx = (1/2)(2)^(-1/2)(2) = 1/√2.

If y = √(x^2 + 1), find dy/dx at x = 1.

If y = √(x^2 + 1), find dy/dx at x = 1.

Correct Answer: 1/√2

Step 1: Start with the equation y = √(x^2 + 1).
Step 2: Rewrite the square root as a power: y = (x^2 + 1)^(1/2).
Step 3: Use the power rule and chain rule to find the derivative dy/dx.
Step 4: The derivative of (x^2 + 1)^(1/2) is (1/2)(x^2 + 1)^(-1/2) * (derivative of x^2 + 1).
Step 5: The derivative of x^2 + 1 is 2x.
Step 6: Combine the results: dy/dx = (1/2)(x^2 + 1)^(-1/2)(2x).
Step 7: Simplify the expression: dy/dx = (x)(x^2 + 1)^(-1/2).
Step 8: Now, substitute x = 1 into the derivative: dy/dx = (1)(1^2 + 1)^(-1/2).
Step 9: Calculate (1^2 + 1) = 2, so dy/dx = (1)(2)^(-1/2).
Step 10: The final result is dy/dx = 1/√2.

Implicit Differentiation – The question tests the ability to differentiate a function defined implicitly, specifically using the chain rule.
Application of Derivatives – It assesses the understanding of how to evaluate the derivative at a specific point.