If y = √(4x^2 + 1), find dy/dx.

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Question: If y = √(4x^2 + 1), find dy/dx.

Options:

Correct Answer: 4x/(√(4x^2 + 1))

Solution:

dy/dx = (4x)/(2√(4x^2 + 1)) = 4x/(√(4x^2 + 1)).

If y = √(4x^2 + 1), find dy/dx.

If y = √(4x^2 + 1), find dy/dx.

Correct Answer: 4x/(√(4x^2 + 1))

Step 1: Start with the equation y = √(4x^2 + 1).
Step 2: Rewrite the square root as an exponent: y = (4x^2 + 1)^(1/2).
Step 3: Use the chain rule to differentiate y with respect to x. The chain rule states that if y = u^n, then dy/dx = n * u^(n-1) * (du/dx).
Step 4: Identify u = 4x^2 + 1 and n = 1/2.
Step 5: Differentiate u with respect to x: du/dx = d(4x^2 + 1)/dx = 8x.
Step 6: Apply the chain rule: dy/dx = (1/2) * (4x^2 + 1)^(1/2 - 1) * (8x).
Step 7: Simplify the expression: dy/dx = (1/2) * (4x^2 + 1)^(-1/2) * (8x).
Step 8: Rewrite (4x^2 + 1)^(-1/2) as 1/√(4x^2 + 1).
Step 9: Combine the terms: dy/dx = (1/2) * 8x / √(4x^2 + 1).
Step 10: Simplify further: dy/dx = 4x / √(4x^2 + 1).

Implicit Differentiation – The question tests the ability to differentiate a function defined in terms of a square root, requiring the application of the chain rule.
Chain Rule – The solution involves using the chain rule to differentiate the outer function (the square root) and the inner function (the polynomial).