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 x/(e^x+1)
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Derivative of (x^2-1)/((2*x))
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Derivative of sqrt(x)^2-2*x
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Derivative of sin(4*x-1)^(3)
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Identical expressions
- (lnx^x)/x^lnx
- (lnx to the power of x) divide by x to the power of lnx
- (lnxx)/xlnx
- lnxx/xlnx
- lnx^x/x^lnx
- (lnx^x) divide by x^lnx
Derivative of (lnx^x)/x^lnx
The solution
You have entered
[src]
x log (x) ------- log(x) x
$$\frac{\log{\left(x \right)}^{x}}{x^{\log{\left(x \right)}}}$$
/ x \ d |log (x)| --|-------| dx| log(x)| \x /
$$\frac{d}{d x} \frac{\log{\left(x \right)}^{x}}{x^{\log{\left(x \right)}}}$$
Detail solution
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Apply the quotient rule, which is:
and .
To find :
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Don't know the steps in finding this derivative.
But the derivative is
To find :
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Don't know the steps in finding this derivative.
But the derivative is
Now plug in to the quotient rule:
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Now simplify:
The answer is:
The first derivative
[src]
-log(x) x
-log(x) x / 1 \ 2*x *log (x)*log(x)
x *log (x)*|------ + log(log(x))| - -------------------------
\log(x) / x
$$x^{- \log{\left(x \right)}} \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right) \log{\left(x \right)}^{x} - \frac{2 x^{- \log{\left(x \right)}} \log{\left(x \right)} \log{\left(x \right)}^{x}}{x}$$
The second derivative
[src]
/ 1 / 1 \ \
| 2 / 2 \ 1 - ------ 4*|------ + log(log(x))|*log(x)|
-log(x) x |/ 1 \ 2*\-1 + 2*log (x) + log(x)/ log(x) \log(x) / |
x *log (x)*||------ + log(log(x))| + --------------------------- + ---------- - -------------------------------|
|\log(x) / 2 x*log(x) x |
\ x /
$$x^{- \log{\left(x \right)}} \left(\left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)^{2} + \frac{1 - \frac{1}{\log{\left(x \right)}}}{x \log{\left(x \right)}} - \frac{4 \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right) \log{\left(x \right)}}{x} + \frac{2 \cdot \left(2 \log{\left(x \right)}^{2} + \log{\left(x \right)} - 1\right)}{x^{2}}\right) \log{\left(x \right)}^{x}$$
The third derivative
[src]
/ / 1 \ \
| 2 | 2 1 - ------| |
| 1 - ------- |/ 1 \ log(x)| / 1 \ / 2 \ / 1 \ / 1 \|
| 3 / 3 2 \ 2 6*||------ + log(log(x))| + ----------|*log(x) 6*|------ + log(log(x))|*\-1 + 2*log (x) + log(x)/ 3*|1 - ------|*|------ + log(log(x))||
-log(x) x |/ 1 \ 2*\-3 - 4*log(x) + 4*log (x) + 6*log (x)/ log (x) \\log(x) / x*log(x) / \log(x) / \ log(x)/ \log(x) /|
x *log (x)*||------ + log(log(x))| - ----------------------------------------- - ----------- - ----------------------------------------------- + -------------------------------------------------- + -------------------------------------|
|\log(x) / 3 2 x 2 x*log(x) |
\ x x *log(x) x /
$$x^{- \log{\left(x \right)}} \left(\left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)^{3} + \frac{3 \cdot \left(1 - \frac{1}{\log{\left(x \right)}}\right) \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)}{x \log{\left(x \right)}} - \frac{6 \left(\left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right)^{2} + \frac{1 - \frac{1}{\log{\left(x \right)}}}{x \log{\left(x \right)}}\right) \log{\left(x \right)}}{x} - \frac{1 - \frac{2}{\log{\left(x \right)}^{2}}}{x^{2} \log{\left(x \right)}} + \frac{6 \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right) \left(2 \log{\left(x \right)}^{2} + \log{\left(x \right)} - 1\right)}{x^{2}} - \frac{2 \cdot \left(4 \log{\left(x \right)}^{3} + 6 \log{\left(x \right)}^{2} - 4 \log{\left(x \right)} - 3\right)}{x^{3}}\right) \log{\left(x \right)}^{x}$$
The graph