Mister Exam
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- Derivative of a implicit defined function
- Derivative of Parametric Function
- Partial derivative of the function
- Curve tracing functions Step by Step
- Integral Step by Step
- Differential equations Step by Step
- Limits Step by Step
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How to use it?
- Derivative of:
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Derivative of e^2*x
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Derivative of x^sin(x)
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Derivative of sqrt(x^2+1)
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Derivative of 4/x^2
- Sum of series:
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(1+1/n)^n
- Limit of the function:
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(1+1/n)^n
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Identical expressions
- (one + one /n)^n
- (1 plus 1 divide by n) to the power of n
- (one plus one divide by n) to the power of n
- (1+1/n)n
- 1+1/nn
- 1+1/n^n
- (1+1 divide by n)^n
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Similar expressions
- (1-1/n)^n
Derivative of (1+1/n)^n
The solution
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
The answer is:
The first derivative
[src]
n
/ 1\ / 1 / 1\\
|1 + -| *|- --------- + log|1 + -||
\ n/ | / 1\ \ n/|
| n*|1 + -| |
\ \ n/ /
$$\left(1 + \frac{1}{n}\right)^{n} \left(\log{\left(1 + \frac{1}{n} \right)} - \frac{1}{n \left(1 + \frac{1}{n}\right)}\right)$$
The second derivative
[src]
n / 2 \
/ 1\ |/ 1 / 1\\ 1 |
|1 + -| *||- --------- + log|1 + -|| - -----------|
\ n/ || / 1\ \ n/| 2|
|| n*|1 + -| | 3 / 1\ |
|\ \ n/ / n *|1 + -| |
\ \ n/ /
$$\left(1 + \frac{1}{n}\right)^{n} \left(\left(\log{\left(1 + \frac{1}{n} \right)} - \frac{1}{n \left(1 + \frac{1}{n}\right)}\right)^{2} - \frac{1}{n^{3} \left(1 + \frac{1}{n}\right)^{2}}\right)$$
The third derivative
[src]
/ / 1 / 1\\\
| 3*|- --------- + log|1 + -|||
| | / 1\ \ n/||
n | 3 | n*|1 + -| ||
/ 1\ |/ 1 / 1\\ 2 3 \ \ n/ /|
|1 + -| *||- --------- + log|1 + -|| - ----------- + ----------- - ----------------------------|
\ n/ || / 1\ \ n/| 3 2 2 |
|| n*|1 + -| | 5 / 1\ 4 / 1\ 3 / 1\ |
|\ \ n/ / n *|1 + -| n *|1 + -| n *|1 + -| |
\ \ n/ \ n/ \ n/ /
$$\left(1 + \frac{1}{n}\right)^{n} \left(\left(\log{\left(1 + \frac{1}{n} \right)} - \frac{1}{n \left(1 + \frac{1}{n}\right)}\right)^{3} - \frac{3 \left(\log{\left(1 + \frac{1}{n} \right)} - \frac{1}{n \left(1 + \frac{1}{n}\right)}\right)}{n^{3} \left(1 + \frac{1}{n}\right)^{2}} + \frac{3}{n^{4} \left(1 + \frac{1}{n}\right)^{2}} - \frac{2}{n^{5} \left(1 + \frac{1}{n}\right)^{3}}\right)$$