Question

If $x^y = \log x$ , then $\frac{dy}{dx} $ at the point where the curve cuts the $x-axis$ is

• A. $e$
• B. $\frac{1}{e}$
• C. $1$
• D. 0

Answer 1

The Correct Option is B

We have, \(x_y = \log x\) ...(i) 
Taking log on both sides of (i), we get 
\(y \log x =\log \left(\log x\right)\Rightarrow y =\frac{\log \left(\log x\right)}{\log x}\) ....(ii) \(\therefore\:\frac{dy}{dx} = \frac{\log x\left(\frac{1}{\log x}\right)\left(\frac{1}{x}\right)-\log \left(\log x\right)\left(\frac{1}{x}\right) -\log \left(\log x\right)\left(\frac{1}{x}\right)}{\left(\log x\right)^{2}}\) 
The point where the curve cuts the x-axis is (e, 0). 
\(\therefore \:\: \frac{dy}{dx}|_{at (e,0)} = \frac{1.1 . \frac{1}{e} - 0}{(1)^2} = \frac{1}{e}\)

Function continuity: This concept makes more sense when described in terms of limitations. If a real function f(x) is close to becoming f(c), then it may be said to be continuous at a position like x = c.

lim_{x -> a} f(x) = f(a)

Conditions for a function's continuity: Any function must fulfil the following requirements in order to be continuous:

• Only if f(a) is a real integer and the function f(x) is stated at x = a, is the function continuous.
• As x gets closer to a, the function has a limit.
• The function's limit as x gets closer to a must match the function's value at x = a.

A function f(x) is differentiable at a point x = a, if f ' (a), i.e., the derivative of the function exists at each point of its domain.

The differentiability of a function is represented as:

f ' (x) = f (x + h) – f(x) / h

If a function f is continuous at any point, the same function is also differentiable at any point x = c in its domain. However, vice versa is not always applicable.