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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 z
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Derivative of x^5*cos(x)
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Derivative of x^(3/4)
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Derivative of 2e^x
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Identical expressions
- arctg((tg(x)-ctg(x))/(two)^ zero , five)
- arctg((tg(x) minus ctg(x)) divide by (2) to the power of 0,5)
- arctg((tg(x) minus ctg(x)) divide by (two) to the power of zero , five)
- arctg((tg(x)-ctg(x))/(2)0,5)
- arctgtgx-ctgx/20,5
- arctgtgx-ctgx/2^0,5
- arctg((tg(x)-ctg(x)) divide by (2)^0,5)
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Similar expressions
- arctg((tg(x)+ctg(x))/(2)^0,5)
Derivative of arctg((tg(x)-ctg(x))/(2)^0,5)
The solution
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[src]
/tan(x) - cot(x)\
atan|---------------|
| ___ |
\ \/ 2 /
$$\operatorname{atan}{\left(\frac{\tan{\left(x \right)} - \cot{\left(x \right)}}{\sqrt{2}} \right)}$$
atan((tan(x) - cot(x))/sqrt(2))
The first derivative
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___ / 2 2 \
\/ 2 *\2 + cot (x) + tan (x)/
-----------------------------
/ 2\
| (tan(x) - cot(x)) |
2*|1 + ------------------|
\ 2 /
$$\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{2 \left(\frac{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2}}{2} + 1\right)}$$
The second derivative
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/ 2 \
| / 2 2 \ |
___ |/ 2 \ / 2 \ \2 + cot (x) + tan (x)/ *(-cot(x) + tan(x))|
2*\/ 2 *|\1 + tan (x)/*tan(x) - \1 + cot (x)/*cot(x) - -------------------------------------------|
| 2 |
\ 2 + (-cot(x) + tan(x)) /
---------------------------------------------------------------------------------------------------
2
2 + (-cot(x) + tan(x))
$$\frac{2 \sqrt{2} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)^{2}}{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 2}\right)}{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 2}$$
The third derivative
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/ 3 3 \
| 2 2 / 2 2 \ 2 / 2 2 \ // 2 \ / 2 \ \ / 2 2 \|
___ |/ 2 \ / 2 \ \2 + cot (x) + tan (x)/ 2 / 2 \ 2 / 2 \ 4*(-cot(x) + tan(x)) *\2 + cot (x) + tan (x)/ 6*(-cot(x) + tan(x))*\\1 + tan (x)/*tan(x) - \1 + cot (x)/*cot(x)/*\2 + cot (x) + tan (x)/|
2*\/ 2 *|\1 + cot (x)/ + \1 + tan (x)/ - ------------------------ + 2*cot (x)*\1 + cot (x)/ + 2*tan (x)*\1 + tan (x)/ + ---------------------------------------------- - ------------------------------------------------------------------------------------------|
| 2 2 2 |
| 2 + (-cot(x) + tan(x)) / 2\ 2 + (-cot(x) + tan(x)) |
\ \2 + (-cot(x) + tan(x)) / /
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2
2 + (-cot(x) + tan(x))
$$\frac{2 \sqrt{2} \left(- \frac{6 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right) \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 2} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)^{3}}{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 2} + \frac{4 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)^{3}}{\left(\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 2\right)^{2}}\right)}{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 2}$$