First slide

Derivatives

Question

$If y^{2} = a x^{2} + b x + c, then y^{3} \frac{d^{2} y}{d x^{2}} is$

Moderate

A

a constant
B

a function of x only
C

a function of y only
D

a function of x and y

Solution

$\begin{array}{l} 2 y \frac{d y}{d x} = 2 a x + b \\ 2 {(\frac{d y}{d x})}^{2} + 2 y \frac{d^{2} y}{d x^{2}} = 2 a \\ y \frac{d^{2} y}{d x^{2}} = a - {(\frac{d y}{d x})}^{2} = a - {(\frac{2 a x + b}{2 y})}^{2} \\ = \frac{4 a y^{2} - (2 a x + b)^{2}}{4 y^{2}} \\ \Rightarrow 4 y^{3} \frac{d^{2} y}{d x^{2}} = 4 a (a x^{2} + b x + c) - (4 a^{2} x^{2} + 4 a b x + b^{2}) \\ \Rightarrow 4 y^{3} \frac{d^{2} y}{d x^{2}} = 4 a c - b^{2} = c o n s t a n t \end{array}$

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