Differential Calculus
Second derivative via parametric differentiation
Grade Class 12
Question:
If $x=\cos\theta$ and $y=\sin^3\theta$, then $\left|y\dfrac{d^2y}{dx^2}+\left(\dfrac{dy}{dx}\right)^2\right|$ at $\theta=\dfrac{\pi}{2}$ is
Step-by-Step Solution
Key Concept: $dy/dx=-3\sin\theta\cos\theta$. $d^2y/dx^2=3\cos2\theta/\sin\theta$. At $\theta=\pi/2$: $y=1$, $dy/dx=0$, $d^2y/dx^2=-3$. Expression $=|1\cdot(-3)+0|=3$.
3.
Correct Answer: 3