Limits, Continuity & Differentiability
Differentiation
nta_abhyas_2025
Grade 12
Question:
If $y = (1+x^2)^n + \sin^{-1}(\sin^2 x)$, then $\frac{d^2y}{dx^2}$ at $x = 0$ is
Step-by-Step Solution
Key Concept: Recognize when to apply logarithmic differentiation for complex products and quotients involving exponentials and logarithms
Putting $x = 0$ and $y = -1$, we get $\frac{dy}{dx} = (1+y)^2\left(\frac{x}{x+1} + \ln(1+x)\right)$. Using the chain rule and logarithmic differentiation techniques, the derivative simplifies to yield $\frac{dy}{dx} = \frac{-\sin 2y}{\sqrt{1-\sin 4y}}$
Correct Answer: 1