Q1467: Differentiate $\left(\frac{5^{x}}{x^{5}}\right)$ with respec...

Class CBSE Class 12 Mathematics Derivatives Q #1467

KNOWLEDGE BASED

APPLY

2 Marks 2025 AISSCE(Board Exam) VSA

Differentiate $\left(\frac{5^{x}}{x^{5}}\right)$ with respect to x.

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AI Tutor Explanation

Detailed Solution

Step 1: Apply Quotient Rule

Let $y = \frac{5^x}{x^5}$. We need to find $\frac{dy}{dx}$. We will use the quotient rule for differentiation, which states that if $y = \frac{u}{v}$, then $\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$. Here, $u = 5^x$ and $v = x^5$.

Step 2: Differentiate u and v

First, we find the derivative of $u = 5^x$ with respect to $x$. $$ \frac{du}{dx} = 5^x \ln{5} $$ Next, we find the derivative of $v = x^5$ with respect to $x$. $$ \frac{dv}{dx} = 5x^4 $$

Step 3: Apply the Quotient Rule Formula

Now, we substitute these derivatives into the quotient rule formula: $$ \frac{dy}{dx} = \frac{x^5(5^x \ln{5}) - 5^x(5x^4)}{(x^5)^2} $$

Step 4: Simplify the Expression

We can simplify the expression by factoring out $5^x x^4$ from the numerator: $$ \frac{dy}{dx} = \frac{5^x x^4 (x \ln{5} - 5)}{x^{10}} $$ Further simplification gives: $$ \frac{dy}{dx} = \frac{5^x (x \ln{5} - 5)}{x^6} $$

Final Answer: $\frac{5^x (x \ln{5} - 5)}{x^6}$

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because the student needs to apply the quotient rule and differentiation formulas to solve the problem.

Knowledge Dimension: PROCEDURAL

Justification: The student needs to know the procedure for differentiating using the quotient rule and the derivatives of exponential and power functions.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of differentiation rules and their application.