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).
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