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 e^x*x
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Derivative of e^(-5*x)
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Derivative of 1/x^6
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Derivative of 2/(x^2-1)
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Identical expressions
- arcsin((sin(x))/(sqrt(one +sin(x)^ two)))
- arc sinus of (( sinus of (x)) divide by ( square root of (1 plus sinus of (x) squared )))
- arc sinus of (( sinus of (x)) divide by ( square root of (one plus sinus of (x) to the power of two)))
- arcsin((sin(x))/(√(1+sin(x)^2)))
- arcsin((sin(x))/(sqrt(1+sin(x)2)))
- arcsinsinx/sqrt1+sinx2
- arcsin((sin(x))/(sqrt(1+sin(x)²)))
- arcsin((sin(x))/(sqrt(1+sin(x) to the power of 2)))
- arcsinsinx/sqrt1+sinx^2
- arcsin((sin(x)) divide by (sqrt(1+sin(x)^2)))
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Similar expressions
- arcsin((sin(x))/(sqrt(1-sin(x)^2)))
- arcsin((sinx)/(sqrt(1+sinx^2)))
Derivative of arcsin((sin(x))/(sqrt(1+sin(x)^2)))
The solution
You have entered
[src]
/ sin(x) \
asin|----------------|
| _____________|
| / 2 |
\\/ 1 + sin (x) /
$$\operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} \right)}$$
asin(sin(x)/sqrt(1 + sin(x)^2))
The first derivative
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2
cos(x) sin (x)*cos(x)
---------------- - ----------------
_____________ 3/2
/ 2 / 2 \
\/ 1 + sin (x) \1 + sin (x)/
-----------------------------------
_________________
/ 2
/ sin (x)
/ 1 - -----------
/ 2
\/ 1 + sin (x)
$$\frac{\frac{\cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} - \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}}}$$
The second derivative
[src]
/ 2 2 2 2 _________________ \
| sin (x) 3*cos (x) 3*cos (x)*sin (x) / 2 |
| 1 - ----------- + ----------- - ----------------- / sin (x) 2 |
| 2 2 2 / 1 - ----------- *cos (x)|
| 1 + sin (x) 1 + sin (x) / 2 \ / 2 |
| \1 + sin (x)/ \/ 1 + sin (x) |
|- ------------------------------------------------- + -------------------------------|*sin(x)
| _________________ 2 |
| / 2 1 + sin (x) |
| / sin (x) |
| / 1 - ----------- |
| / 2 |
\ \/ 1 + sin (x) /
----------------------------------------------------------------------------------------------
_____________
/ 2
\/ 1 + sin (x)
$$\frac{\left(\frac{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}}}\right) \sin{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}$$
The third derivative
[src]
/ 2 2 4 2 2 2 4 4 2 2 2 4 / 2 2 2 2 \ _________________ \
| 10*sin (x) 3*cos (x) 9*sin (x) 18*cos (x)*sin (x) 15*cos (x)*sin (x) 2 2 sin (x) 5*cos (x)*sin (x) 4*cos (x)*sin (x) 2 | sin (x) 3*cos (x) 3*cos (x)*sin (x)| / 2 |
| 1 - ----------- + ----------- + -------------- - ------------------ + ------------------ cos (x) - sin (x) + ----------- - ----------------- + ----------------- 2*sin (x)*|1 - ----------- + ----------- - -----------------| / sin (x) 2 2 |
| 2 2 2 2 3 2 2 2 | 2 2 2 | 3* / 1 - ----------- *cos (x)*sin (x)|
| 1 + sin (x) 1 + sin (x) / 2 \ / 2 \ / 2 \ 1 + sin (x) 1 + sin (x) / 2 \ | 1 + sin (x) 1 + sin (x) / 2 \ | / 2 |
| \1 + sin (x)/ \1 + sin (x)/ \1 + sin (x)/ \1 + sin (x)/ \ \1 + sin (x)/ / \/ 1 + sin (x) |
|- ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------- - ------------------------------------------------------------- + -----------------------------------------|*cos(x)
| _________________ _________________ _________________ 2 |
| / 2 / 2 / 2 / 2 \ |
| / sin (x) / 2 \ / sin (x) / 2 \ / sin (x) \1 + sin (x)/ |
| / 1 - ----------- \1 + sin (x)/* / 1 - ----------- \1 + sin (x)/* / 1 - ----------- |
| / 2 / 2 / 2 |
\ \/ 1 + sin (x) \/ 1 + sin (x) \/ 1 + sin (x) /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
_____________
/ 2
\/ 1 + sin (x)
$$\frac{\left(\frac{3 \sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} - \frac{1 - \frac{10 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{9 \sin^{4}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} - \frac{18 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} + \frac{15 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{3}}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}}} - \frac{2 \left(1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \sin^{2}{\left(x \right)}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}} \left(\sin^{2}{\left(x \right)} + 1\right)} + \frac{- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{\sin^{4}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{5 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{4 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}} \left(\sin^{2}{\left(x \right)} + 1\right)}\right) \cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}$$