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Logarithmic differentiation is the natural method for a function of the form

  • A.\( y = u(x) + v(x) \)
  • B.\( y = u(x)/v(x) \) with polynomials only
  • C.\( y = \sin u(x) \) with \( u \) linear
  • D.\( y = u(x)^{v(x)} \)

Answer: D

When both the base and the exponent vary, neither the power rule nor the exponential rule applies. Taking logarithms gives \( \ln y = v \ln u \), which differentiates by the product rule, and multiplying back by \( y \) recovers the derivative. Sums, quotients of polynomials and composed trigonometric functions all yield to the ordinary rules

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