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 x^7*e^x
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Derivative of 1/(x+1)^2
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Derivative of sin(4x)
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Derivative of 10x
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Identical expressions
- arctg((tgx-ctgx)/sqrt(two)(two))
- arctg((tgx minus ctgx) divide by square root of (2)(2))
- arctg((tgx minus ctgx) divide by square root of (two)(two))
- arctg((tgx-ctgx)/√(2)(2))
- arctgtgx-ctgx/sqrt22
- arctg((tgx-ctgx) divide by sqrt(2)(2))
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Similar expressions
- arctg((tgx+ctgx)/sqrt(2)(2))
Derivative of arctg((tgx-ctgx)/sqrt(2)(2))
The solution
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[src]
/tan(x) - cot(x) \
atan|---------------*2|
| ___ |
\ \/ 2 /
$$\operatorname{atan}{\left(2 \frac{\tan{\left(x \right)} - \cot{\left(x \right)}}{\sqrt{2}} \right)}$$
atan(((tan(x) - cot(x))/sqrt(2))*2)
The first derivative
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___ / 2 2 \
\/ 2 *\2 + cot (x) + tan (x)/
-----------------------------
2
1 + 2*(tan(x) - cot(x))
$$\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1}$$
The second derivative
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/ 2 \
| / 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 |
\ 1 + 2*(-cot(x) + tan(x)) /
-----------------------------------------------------------------------------------------------------
2
1 + 2*(-cot(x) + tan(x))
$$\frac{2 \sqrt{2} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)^{2}}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1}\right)}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1}$$
The third derivative
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/ 3 3 \
| 2 2 / 2 2 \ 2 / 2 2 \ // 2 \ / 2 \ \ / 2 2 \|
___ |/ 2 \ / 2 \ 2*\2 + cot (x) + tan (x)/ 2 / 2 \ 2 / 2 \ 16*(-cot(x) + tan(x)) *\2 + cot (x) + tan (x)/ 12*(-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 |
| 1 + 2*(-cot(x) + tan(x)) / 2\ 1 + 2*(-cot(x) + tan(x)) |
\ \1 + 2*(-cot(x) + tan(x)) / /
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2
1 + 2*(-cot(x) + tan(x))
$$\frac{2 \sqrt{2} \left(- \frac{12 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right) \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)^{3}}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1} + \frac{16 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)^{3}}{\left(2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1\right)^{2}}\right)}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1}$$