Mister Exam
Other calculators
- 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 (x-3)^4
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Derivative of (x+3)^3
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Derivative of (x^2-1)^3
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Derivative of (x+1)/(x^2+1)
- Limit of the function:
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(1/x)^(1/x)
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Identical expressions
- (one /x)^(one /x)
- (1 divide by x) to the power of (1 divide by x)
- (one divide by x) to the power of (one divide by x)
- (1/x)(1/x)
- 1/x1/x
- 1/x^1/x
- (1 divide by x)^(1 divide by x)
Derivative of (1/x)^(1/x)
The solution
You have entered
[src]
_____ / 1 x / 1*- \/ x
$$\left(1 \cdot \frac{1}{x}\right)^{1 \cdot \frac{1}{x}}$$
/ _____\ d | / 1 | --|x / 1*- | dx\\/ x /
$$\frac{d}{d x} \left(1 \cdot \frac{1}{x}\right)^{1 \cdot \frac{1}{x}}$$
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
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Now simplify:
The answer is:
The first derivative
[src]
/ / 1\\
_____ | log|1*-||
/ 1 | 1 \ x/|
x / 1*- *|- -- - --------|
\/ x | 2 2 |
\ x x /
$$\left(1 \cdot \frac{1}{x}\right)^{\frac{1}{x}} \left(- \frac{\log{\left(1 \cdot \frac{1}{x} \right)}}{x^{2}} - \frac{1}{x^{2}}\right)$$
The second derivative
[src]
/ 2\
| / /1\\ |
___ | |1 + log|-|| |
/ 1 | /1\ \ \x// |
x / - *|3 + 2*log|-| + -------------|
\/ x \ \x/ x /
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3
x
$$\frac{\left(2 \log{\left(\frac{1}{x} \right)} + \frac{\left(\log{\left(\frac{1}{x} \right)} + 1\right)^{2}}{x} + 3\right) \left(\frac{1}{x}\right)^{\frac{1}{x}}}{x^{3}}$$
The third derivative
[src]
/ 3 \
| / /1\\ / /1\\ / /1\\|
___ | |1 + log|-|| 3*|1 + log|-||*|3 + 2*log|-|||
/ 1 | /1\ \ \x// \ \x// \ \x//|
-x / - *|11 + 6*log|-| + ------------- + -----------------------------|
\/ x | \x/ 2 x |
\ x /
-------------------------------------------------------------------------
4
x
$$- \frac{\left(6 \log{\left(\frac{1}{x} \right)} + \frac{3 \left(\log{\left(\frac{1}{x} \right)} + 1\right) \left(2 \log{\left(\frac{1}{x} \right)} + 3\right)}{x} + \frac{\left(\log{\left(\frac{1}{x} \right)} + 1\right)^{3}}{x^{2}} + 11\right) \left(\frac{1}{x}\right)^{\frac{1}{x}}}{x^{4}}$$
The graph