Limits, Continuity & Differentiability
Differentiation
nta_abhyas_2025
Grade None
Question:
If $y = \log_a 2 + \log_c 10 + \log_c 10 + \log_{10} 10$, then $\frac{dy}{dx}$ is equal to
\frac{\log 10}{\log a}
\frac{1}{\log 10} + \frac{1}{\log a}
\frac{1}{\log 10} + \frac{\log 10}{x \log a}
None of these
Step-by-Step Solution
Key Concept: Differentiation of logarithmic functions using chain rule and quotient rule
We have $y = \log_e x + \log_e 10 + \log_e z + \log_e 10 = \log_e u + \log_e 10 + 1 + 1$. Taking the derivative with respect to $x$: $\frac{dy}{dx} = \frac{1}{\log_e x} \cdot \frac{\log_e 10}{x\log_e x^2}$. Simplifying gives $\frac{dy}{dx} = \frac{1}{\log_e u} - \frac{\log_e 10}{x\log_e x^2}$.
Correct Answer: 1