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Leibniz's rule for differentiating \( \displaystyle\int_{a}^{b} f(x,t)\,dx \) with respect to \( t \), the limits being constant, gives

  • A.\( \displaystyle\int_{a}^{b} \frac{\partial f}{\partial t}\,dx \)
  • B.\( \frac{\partial}{\partial t}\left[f(b,t) - f(a,t)\right] \)
  • C.\( \displaystyle\int_{a}^{b} \frac{\partial f}{\partial x}\,dx \)
  • D.\( f(b,t)\,\frac{db}{dt} - f(a,t)\,\frac{da}{dt} \)

Answer: A

With fixed limits the derivative passes inside the integral, so

\[ \frac{d}{dt}\int_{a}^{b} f(x,t)\,dx \;=\; \int_{a}^{b} \frac{\partial f}{\partial t}\,dx. \]

The boundary terms in the last form appear only when the limits themselves depend on \( t \), and differentiating with respect to \( x \) instead would undo the integration

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