Question

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find the derivative of log(x+√(x^2+a^2)​)​

Answer 1

Assuming that log is the natural logarithm:

y = log[x + √[x² + a²]]

y' = d/dx[log[x + √[x² + a²]]] --- {Needs a chain rule}

y' = 1 / [x + √[x² + a²]] * d/dx[x + √[x² + a²]]

y' = 1 / [x + √[x² + a²]] * (1 + dy/dx[√[x² + a²]]) --- {Need to chain rule one more time}

y' = 1 / [x + √[x² + a²]] * (1 + 1 / 2√[x² + a²] * dy/dx[x² + a²])

y' = 1 / [x + √[x² + a²]] * (1 + 1 / 2√[x² + a²] * 2x)

y' = (1 + x) ÷ √[x² + a²][x + √[x² + a²]]

Answer 2

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