Limits, Continuity & Differentiability
Differentiation
nta_abhyas_2025
Grade 12
Question:
If $I_n = \frac{d^n}{dx^n}(x^n\ln x)$, then the value of $\frac{d}{dx}(I_7 - 7I_6)$ is equal to
Step-by-Step Solution
Key Concept: Recurrence relations for integrals involving exponential and power functions
We have the recurrence relation $I_n = \frac{e^x}{n}(x^n + nx^{n-1}\ln x)$ and $I_n = (n-1)!+a I_{n-1}$ where $a_k = (n-1)!$. Putting $n = 7$: $I_7 - 7I_6 = 6! = 720$. This follows from the given recurrence relation for the integral involving exponential and logarithmic terms.
Correct Answer: 720