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- Derivative of a implicit defined function
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- Partial derivative of the function
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How to use it?
- Derivative of:
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Derivative of sin(x/3)^(2)*cot(x/2)
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Derivative of atan(sqrt(x))
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Derivative of atan(2*x)
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Derivative of cos(2*x-3)
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Identical expressions
- arcsin((sinx)/(one +sin^2x))
- arc sinus of (( sinus of x) divide by (1 plus sinus of squared x))
- arc sinus of (( sinus of x) divide by (one plus sinus of squared x))
- arcsin((sinx)/(1+sin2x))
- arcsinsinx/1+sin2x
- arcsin((sinx)/(1+sin²x))
- arcsin((sinx)/(1+sin to the power of 2x))
- arcsinsinx/1+sin^2x
- arcsin((sinx) divide by (1+sin^2x))
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Similar expressions
- arcsin((sinx)/(1-sin^2x))
Derivative of arcsin((sinx)/(1+sin^2x))
The solution
You have entered
[src]
/ sin(x) \
asin|-----------|
| 2 |
\1 + sin (x)/
$$\operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} \right)}$$
asin(sin(x)/(1 + sin(x)^2))
The first derivative
[src]
2
cos(x) 2*sin (x)*cos(x)
----------- - ----------------
2 2
1 + sin (x) / 2 \
\1 + sin (x)/
------------------------------
____________________
/ 2
/ sin (x)
/ 1 - --------------
/ 2
/ / 2 \
\/ \1 + sin (x)/
$$\frac{\frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}}}$$
The second derivative
[src]
/ 2 \
| / 2 \ |
| | 2*sin (x) | 2 |
| |1 - -----------| *cos (x) |
| 2 2 2 2 | 2 | |
| 6*cos (x) 2*sin (x) 8*cos (x)*sin (x) \ 1 + sin (x)/ |
|-1 - ----------- + ----------- + ----------------- + -----------------------------------|*sin(x)
| 2 2 2 2 / 2 \|
| 1 + sin (x) 1 + sin (x) / 2 \ / 2 \ | sin (x) ||
| \1 + sin (x)/ \1 + sin (x)/ *|1 - --------------||
| | 2||
| | / 2 \ ||
\ \ \1 + sin (x)/ //
-------------------------------------------------------------------------------------------------
____________________
/ 2
/ 2 \ / sin (x)
\1 + sin (x)/* / 1 - --------------
/ 2
/ / 2 \
\/ \1 + sin (x)/
$$\frac{\left(-1 + \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{8 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} + \frac{\left(1 - \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}\right)^{2} \cos^{2}{\left(x \right)}}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}} \left(\sin^{2}{\left(x \right)} + 1\right)}$$
The third derivative
[src]
/ / 2 \ / 4 2 2 2 4 \ / 2 \ / 2 2 2 2 \ 3 \
| | 2*sin (x) | | 2 2 2*sin (x) 10*cos (x)*sin (x) 12*cos (x)*sin (x)| 2 | 2*sin (x) | | 2*sin (x) 6*cos (x) 8*cos (x)*sin (x)| / 2 \ |
| |1 - -----------|*|cos (x) - sin (x) + ----------- - ------------------ + ------------------| 2*sin (x)*|1 - -----------|*|1 - ----------- + ----------- - -----------------| | 2*sin (x) | 2 2 |
| | 2 | | 2 2 2 | | 2 | | 2 2 2 | 3*|1 - -----------| *cos (x)*sin (x)|
| 4 2 2 2 4 2 2 \ 1 + sin (x)/ | 1 + sin (x) 1 + sin (x) / 2 \ | \ 1 + sin (x)/ | 1 + sin (x) 1 + sin (x) / 2 \ | | 2 | |
| 24*sin (x) 6*cos (x) 20*sin (x) 48*cos (x)*sin (x) 48*cos (x)*sin (x) \ \1 + sin (x)/ / \ \1 + sin (x)/ / \ 1 + sin (x)/ |
|-1 - -------------- - ----------- + ----------- - ------------------ + ------------------ + --------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------- + ------------------------------------|*cos(x)
| 2 2 2 3 2 2 / 2 \ 2 / 2 \ 2|
| / 2 \ 1 + sin (x) 1 + sin (x) / 2 \ / 2 \ / 2 \ | sin (x) | / 2 \ | sin (x) | 4 / 2 \ |
| \1 + sin (x)/ \1 + sin (x)/ \1 + sin (x)/ \1 + sin (x)/ *|1 - --------------| \1 + sin (x)/ *|1 - --------------| / 2 \ | sin (x) | |
| | 2| | 2| \1 + sin (x)/ *|1 - --------------| |
| | / 2 \ | | / 2 \ | | 2| |
| \ \1 + sin (x)/ / \ \1 + sin (x)/ / | / 2 \ | |
\ \ \1 + sin (x)/ / /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
____________________
/ 2
/ 2 \ / sin (x)
\1 + sin (x)/* / 1 - --------------
/ 2
/ / 2 \
\/ \1 + sin (x)/
$$\frac{\left(-1 + \frac{20 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{24 \sin^{4}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} + \frac{48 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} - \frac{48 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{3}} - \frac{2 \left(1 - \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}\right) \left(1 - \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{8 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \sin^{2}{\left(x \right)}}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \left(\sin^{2}{\left(x \right)} + 1\right)^{2}} + \frac{\left(1 - \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}\right) \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{2 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{10 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{12 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right)}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \left(\sin^{2}{\left(x \right)} + 1\right)^{2}} + \frac{3 \left(1 - \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}\right)^{3} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right)^{2} \left(\sin^{2}{\left(x \right)} + 1\right)^{4}}\right) \cos{\left(x \right)}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}} \left(\sin^{2}{\left(x \right)} + 1\right)}$$
The graph