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y=ln⁡(x)y = \ln(\sqrt{x}). Qual o valor da 2a2^a derivada quando x=1x=1? Ou seja, calcule y′′(1)y''(1)

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Find derivatives using the quotient rule Iright chevron icon

Full solution

Q. y=ln⁡(x)y = \ln(\sqrt{x}). Qual o valor da 2a2^a derivada quando x=1x=1? Ou seja, calcule y′′(1)y''(1)
  1. Rewrite using properties of logarithms: Rewrite the function using properties of logarithms. ln⁡(x)=12ln⁡(x)\ln(\sqrt{x}) = \frac{1}{2}\ln(x).
  2. Find first derivative using chain rule: Find the first derivative y′y' using the chain rule. y′=(12)(1x)=12xy' = (\frac{1}{2})(\frac{1}{x}) = \frac{1}{2x}.
  3. Find second derivative: Now, find the second derivative y′′y''. y′′=−12x2y'' = -\frac{1}{2x^2}.
  4. Evaluate at x=1x=1: Evaluate y′′y'' at x=1x=1. y′′(1)=−12⋅12=−12y''(1) = -\frac{1}{2\cdot1^2} = -\frac{1}{2}.

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