N-th derivative practice sum

Understand the Problem

The question is asking for practice problems related to the n-th derivative, which involves the mathematical concept of taking derivatives multiple times. This typically falls under calculus or mathematical analysis.

Answer

The n-th derivative of $f(x) = e^x$ is $f^{(n)}(x) = e^x$, and for $f(x) = x^n$, it is $f^{(n)}(x) = n!$.
Answer for screen readers

The general form for the n-th derivative of $f(x) = e^x$ is $f^{(n)}(x) = e^x$.

For $f(x) = x^n$, the n-th derivative is 0 when $k > n$ and $f^{(n)}(x) = n!$ when $k = n$.

Steps to Solve

  1. Identify the function to differentiate

Start by determining which function you will be differentiating to find the n-th derivative. For example, a common function is $f(x) = e^x$.

  1. Differentiate the function and identify a pattern

Calculate the first few derivatives of the function. For our example:

  • First derivative: $f'(x) = e^x$
  • Second derivative: $f''(x) = e^x$
  • Third derivative: $f'''(x) = e^x$

Notice that the pattern shows that the derivatives of $e^x$ remain $e^x$.

  1. Generalize the n-th derivative

For the function $f(x) = e^x$, the n-th derivative can be expressed as:

$$ f^{(n)}(x) = e^x $$

  1. Practice problems

Choose other functions such as $f(x) = x^n$ or $f(x) = \sin(x)$.

  • For $f(x) = x^n$, the n-th derivative can be found using the formula:

$$ f^{(n)}(x) = \frac{n!}{(n-k)!} x^{n-k} $$

where $k$ is the order of the derivative.

  1. Solve for n-th derivatives of other functions

Keep practicing by calculating n-th derivatives for various functions, looking for patterns and generalizing.

The general form for the n-th derivative of $f(x) = e^x$ is $f^{(n)}(x) = e^x$.

For $f(x) = x^n$, the n-th derivative is 0 when $k > n$ and $f^{(n)}(x) = n!$ when $k = n$.

More Information

Understanding n-th derivatives is important in calculus, as it reveals how functions behave under continuous change. The n-th derivative indicates the rate of change of a function’s rate of change, tracing deeper into the function's structure.

Tips

  • Forgetting that the n-th derivative of $e^x$ is always $e^x$ regardless of the order of the derivative.
  • Confusing the n-th derivative formula for polynomial functions, which results in incorrect application of factorial next to power terms.

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