Mister Exam
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- Derivative of a implicit defined function
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How to use it?
- Derivative of:
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Derivative of sqrt(x-1)/(sqrt(x^2-x)-1)
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Derivative of 5*x^4-7*x^2-x
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Derivative of 3x-x^2
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Derivative of 4*sqrt(3)*x/3
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Identical expressions
- y=tg(x)^ctg(x)
- y equally tg(x) to the power of ctg(x)
- y=tg(x)ctg(x)
- y=tgxctgx
- y=tgx^ctgx
Derivative of y=tg(x)^ctg(x)
The solution
You have entered
[src]
cot(x) tan (x)
$$\tan^{\cot{\left(x \right)}}{\left(x \right)}$$
tan(x)^cot(x)
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]
/ / 2 \ \
cot(x) |/ 2 \ \1 + tan (x)/*cot(x)|
tan (x)*|\-1 - cot (x)/*log(tan(x)) + --------------------|
\ tan(x) /
$$\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} + \left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(\tan{\left(x \right)} \right)}\right) \tan^{\cot{\left(x \right)}}{\left(x \right)}$$
The second derivative
[src]
/ 2 2 \
|/ / 2 \ \ / 2 \ / 2 \ / 2 \ |
cot(x) || / 2 \ \1 + tan (x)/*cot(x)| / 2 \ \1 + tan (x)/ *cot(x) 2*\1 + cot (x)/*\1 + tan (x)/ / 2 \ |
tan (x)*||- \1 + cot (x)/*log(tan(x)) + --------------------| + 2*\1 + tan (x)/*cot(x) - --------------------- - ----------------------------- + 2*\1 + cot (x)/*cot(x)*log(tan(x))|
|\ tan(x) / 2 tan(x) |
\ tan (x) /
$$\left(\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}\right)^{2} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cot{\left(x \right)}}{\tan^{2}{\left(x \right)}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)} \cot{\left(x \right)}\right) \tan^{\cot{\left(x \right)}}{\left(x \right)}$$
The third derivative
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/ 3 / 2 \ 2 3 2 \
|/ / 2 \ \ / / 2 \ \ | / 2 \ / 2 \ / 2 \| 2 / 2 \ / 2 \ / 2 \ / 2 \ / 2 \ / 2 \ |
cot(x) || / 2 \ \1 + tan (x)/*cot(x)| / 2 \ / 2 \ | / 2 \ \1 + tan (x)/*cot(x)| | / 2 \ \1 + tan (x)/ *cot(x) / 2 \ 2*\1 + cot (x)/*\1 + tan (x)/| / 2 \ 4*\1 + tan (x)/ *cot(x) 2 / 2 \ 2*\1 + tan (x)/ *cot(x) 3*\1 + tan (x)/ *\1 + cot (x)/ / 2 \ 6*\1 + cot (x)/*\1 + tan (x)/*cot(x)|
tan (x)*||- \1 + cot (x)/*log(tan(x)) + --------------------| - 6*\1 + cot (x)/*\1 + tan (x)/ - 3*|- \1 + cot (x)/*log(tan(x)) + --------------------|*|- 2*\1 + tan (x)/*cot(x) + --------------------- - 2*\1 + cot (x)/*cot(x)*log(tan(x)) + -----------------------------| - 2*\1 + cot (x)/ *log(tan(x)) - ----------------------- - 4*cot (x)*\1 + cot (x)/*log(tan(x)) + ----------------------- + ------------------------------ + 4*\1 + tan (x)/*cot(x)*tan(x) + ------------------------------------|
|\ tan(x) / \ tan(x) / | 2 tan(x) | tan(x) 3 2 tan(x) |
\ \ tan (x) / tan (x) tan (x) /
$$\left(\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}\right)^{3} - 3 \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} - \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}\right) \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cot{\left(x \right)}}{\tan^{2}{\left(x \right)}} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)} \cot{\left(x \right)}\right) + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \cot{\left(x \right)}}{\tan^{3}{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cot{\left(x \right)}}{\tan{\left(x \right)}} - 6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan{\left(x \right)}} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)} - 2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} \right)} - 4 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)} \cot^{2}{\left(x \right)}\right) \tan^{\cot{\left(x \right)}}{\left(x \right)}$$
The graph