List-I | List-II |
---|---|
The derivative of \( \log_e x \) with respect to \( \frac{1}{x} \) at \( x = 5 \) is | (I) -5 |
If \( x^3 + x^2y + xy^2 - 21x = 0 \), then \( \frac{dy}{dx} \) at \( (1, 1) \) is | (II) -6 |
If \( f(x) = x^3 \log_e \frac{1}{x} \), then \( f'(1) + f''(1) \) is | (III) 5 |
If \( y = f(x^2) \) and \( f'(x) = e^{\sqrt{x}} \), then \( \frac{dy}{dx} \) at \( x = 0 \) is | (IV) 0 |