If y = sqrt(x^2 + 1), find dy/dx at x = 0.

₹0.0

Question: If y = sqrt(x^2 + 1), find dy/dx at x = 0.

Options:

Correct Answer: 1

Solution:

dy/dx = (1/2)(x^2 + 1)^(-1/2)(2x). At x = 0, dy/dx = (1/2)(1)^(-1/2)(0) = 0.

If y = sqrt(x^2 + 1), find dy/dx at x = 0.

If y = sqrt(x^2 + 1), find dy/dx at x = 0.

Correct Answer: 0

Step 1: Start with the equation y = sqrt(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, so we have dy/dx = (1/2)(x^2 + 1)^(-1/2)(2x).
Step 6: Simplify the expression: dy/dx = (x)(x^2 + 1)^(-1/2).
Step 7: Now, substitute x = 0 into the derivative: dy/dx = (0)(1)^(-1/2).
Step 8: Calculate the result: dy/dx = 0.

Implicit Differentiation – The process of finding the derivative of a function defined implicitly, in this case using the chain rule.
Evaluating Derivatives at a Point – Substituting a specific value into the derivative to find the slope of the tangent line at that point.