Limits, Continuity & Differentiability
Differentiation
nta_abhyas_2025
Grade 12
Question:
If $\log_{10}\left(\frac{x^2}{y}\right) = 2$, then $\frac{dy}{dx} =$
$\frac{x}{5}$
$\frac{y}{5}$
$\frac{x}{4}$
$\frac{y}{4}$
Step-by-Step Solution
Key Concept: Componendo and dividendo theorem for solving logarithmic equations involving ratios
Given $\log_{10}\left(\frac{x+y}{x-y}\right) = 2$, we have $\frac{x+y}{x-y} = (10)^2 = 100$. Applying componendo and dividendo: $\frac{2x}{2y} = \frac{100+1}{100-1} = \frac{101}{99}$, so $\frac{x}{y} = \frac{101}{99}$. Therefore $y = \left(\frac{99}{101}\right)^k x$.
Correct Answer: 4