Finds its nth derivative of question number 6...........................................//...../

To find the nth derivative of a function, we first need to identify the function itself. Since you mentioned "question number 6," I don't have the specific function you're referring to. However, I can guide you through the general process of finding the nth derivative, which can be applied to various functions. Let's break it down step by step.

Understanding Derivatives

The derivative of a function measures how the function's output changes as its input changes. The first derivative gives us the slope of the function, while the second derivative provides information about the curvature. As we continue to take derivatives, we can analyze the behavior of the function more deeply.

Finding the nth Derivative

To find the nth derivative, we can use a few different methods depending on the type of function. Here are some common approaches:

• Polynomial Functions: For a polynomial of the form f(x) = ax^n, the nth derivative can be calculated using the formula:

f^(n)(x) = a * n! / (n-k)! * x^(n-k)

• Exponential Functions: For functions like f(x) = e^(kx), the nth derivative is simply k^n * e^(kx).
• Trigonometric Functions: The derivatives of sine and cosine functions follow a cyclical pattern. For example, the nth derivative of sin(x) is:

f^(n)(x) = sin(x + n * π/2)

Example: nth Derivative of a Polynomial

Let’s say we have a polynomial function:

f(x) = 3x^4 + 2x^3 - x + 5

To find the nth derivative, we can differentiate step by step:

• First derivative: f'(x) = 12x^3 + 6x^2 - 1
• Second derivative: f''(x) = 36x^2 + 12x
• Third derivative: f'''(x) = 72x + 12
• Fourth derivative: f''''(x) = 72
• Fifth derivative and beyond: f^(n)(x) = 0 for n > 4

In this case, after the fourth derivative, all subsequent derivatives are zero, which is a common occurrence with polynomial functions.

Using the General Formula

For polynomials, you can also use the general formula for the nth derivative:

f^(n)(x) = 0 for n greater than the degree of the polynomial.

Practice Makes Perfect

To master finding the nth derivative, practice with different types of functions. Try applying the steps outlined above to exponential and trigonometric functions as well. Each function type has its unique characteristics, and understanding these will enhance your calculus skills.

If you have a specific function from your question number 6, feel free to share it, and I can help you find its nth derivative directly!