Class CBSE Class 12 Mathematics Derivatives Q #1245
KNOWLEDGE BASED
UNDERSTAND
2 Marks 2024 AISSCE(Board Exam) VSA
If $x=e^{x/y}$, prove that $\frac{dy}{dx}=\frac{log~x-1}{(log~x)^{2}}$
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Detailed Solution

Step 1: Take the natural logarithm of both sides

Given the equation $x = e^{x/y}$, we take the natural logarithm (ln) of both sides to simplify the exponential term.

Step 2: Simplify the equation using logarithm properties

Using the property $\ln(e^a) = a$, we get: $\ln(x) = \ln(e^{x/y}) = \frac{x}{y}$

Step 3: Solve for y in terms of x

From the equation $\ln(x) = \frac{x}{y}$, we solve for $y$: $y = \frac{x}{\ln(x)}$

Step 4: Differentiate y with respect to x

Now, we differentiate $y$ with respect to $x$ using the quotient rule, which states that if $y = \frac{u}{v}$, then $\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$. Here, $u = x$ and $v = \ln(x)$.

Step 5: Apply the quotient rule

We have $\frac{du}{dx} = 1$ and $\frac{dv}{dx} = \frac{1}{x}$. Applying the quotient rule: $\frac{dy}{dx} = \frac{\ln(x) \cdot 1 - x \cdot \frac{1}{x}}{(\ln(x))^2} = \frac{\ln(x) - 1}{(\ln(x))^2}$

Step 6: Rewrite using log notation

Since $\ln(x)$ is the same as $\log_e(x)$, we can rewrite the derivative as: $\frac{dy}{dx} = \frac{\log x - 1}{(\log x)^2}$

Final Answer: $\frac{dy}{dx}=\frac{log~x-1}{(log~x)^{2}}$

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the properties of logarithms and differentiation rules to solve the problem.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a series of steps, including taking logarithms, algebraic manipulation, and differentiation using the quotient rule. These are all procedural skills.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of logarithmic differentiation and application of the quotient rule, which are standard topics in the syllabus.

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