If \(y =\sqrt{ax + b}\), then \(y\displaystyle\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2=\)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0
Squaring first removes the root:
\(y^2 = ax + b\)
Differentiating with respect to x:
\( \displaystyle 2y\frac{dy}{dx} = a \)
Differentiating again, by the product rule (a is a constant):
\( \displaystyle 2\left(\frac{dy}{dx}\right)^2 + 2y\frac{d^2y}{dx^2} = 0 \)
Dividing by 2:
\( \displaystyle y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 0 \)