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