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I can see the image you uploaded with a set of mathematical questions from an assignment. Let’s go through some of these questions. If you’d like me to solve a specific one, please let me know.

### Here are the questions listed from the image:

#### Short Answer Questions:
1. Find the \( n^{th} \) differentiation of \( a^x \cos x \).
2. Find \( y_n \), if \( y = \tan^{-1} \left(\frac{x}{c}\right) \).
3. Find \( y_n \), if \( y = x^2 \sin x \) at \( x = 0 \).
4. Find \( y_n \), if \( e^{ax} \sin^2 x \sin 2x \).
5. Examine if Rolle’s theorem is applicable for \( f(x) = \sec x \) in \( [0, 2\pi] \).
6. Expand \( \ln x \) in powers of \( (x - 1) \) up to the third degree term.

#### Long Answer Questions:
1. Expand \( 4x^2 + 7x + 5 \) in powers of \( (x - 3) \).
2. If \( y = \ln \left( x + \sqrt{1 + x^2} \right)^2 \), prove that \( (1 + x^2)y_{n+2} + (2n+1)xy_{n+1} + n^2 y_n = 0 \).
3. Verify Lagrange’s mean value theorem for:
   - \( \sqrt{x^2 - 4} \) in \( [2, 4] \)
   - \( \ln x \) in \( [1, e] \)
4. If \( y = x^{n-1} \ln x \), show that \( y_n = \frac{(n-1)!}{x} \).
5. If \( \cos^{-1} \frac{y}{b} = \log \left( \frac{x}{n} \right)^n \), show that \( x^2 y_{n+2} + (2n+1) xy_{n+1} + 2n^2 y_n = 0 \).
6. If \( y = a \cos \ln x + b \sin \ln x \), show that \( x^2 y_{n+2} + (2n+1) xy_{n+1} + (n^2+1) y_n = 0 \).
7. If \( y = \sin (m \sin^{-1} x) \), show that \( (1 - x^2) y_{n+2} - (2n+1) xy_{n+1} + (m^2 - n^2) y_n = 0 \).

Please let me know which problem you'd like to explore, or if you'd like help with multiple questions!

Find nth differentiation of ax cos x, and other calculus-related problems.

Explore a series of calculus problems involving advanced differentiation techniques, Rolle's Theorem, and Lagrange's Mean Value Theorem. This assignment includes problems like finding the nth differentiation of ax cos x, expanding logarithmic functions, and verifying classical theorems.

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