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- Derivative of:
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Derivative of sin(x)^2
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Derivative of e^(2-x)
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Derivative of cos(x/3)
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Derivative of √x
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Identical expressions
- arccos((seven + four *cos(x))/(four + seven *cos(x)))
- arc co sinus of e of ((7 plus 4 multiply by co sinus of e of (x)) divide by (4 plus 7 multiply by co sinus of e of (x)))
- arc co sinus of e of ((seven plus four multiply by co sinus of e of (x)) divide by (four plus seven multiply by co sinus of e of (x)))
- arccos((7+4cos(x))/(4+7cos(x)))
- arccos7+4cosx/4+7cosx
- arccos((7+4*cos(x)) divide by (4+7*cos(x)))
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Similar expressions
- arccos((7+4*cos(x))/(4-7*cos(x)))
- arccos((7-4*cos(x))/(4+7*cos(x)))
- arccos((7+4*cosx)/(4+7*cosx))
Derivative of arccos((7+4*cos(x))/(4+7*cos(x)))
The solution
You have entered
[src]
/7 + 4*cos(x)\
acos|------------|
\4 + 7*cos(x)/
$$\operatorname{acos}{\left(\frac{4 \cos{\left(x \right)} + 7}{7 \cos{\left(x \right)} + 4} \right)}$$
acos((7 + 4*cos(x))/(4 + 7*cos(x)))
The first derivative
[src]
/ 4*sin(x) 7*(7 + 4*cos(x))*sin(x)\
-|- ------------ + -----------------------|
| 4 + 7*cos(x) 2 |
\ (4 + 7*cos(x)) /
--------------------------------------------
_____________________
/ 2
/ (7 + 4*cos(x))
/ 1 - ---------------
/ 2
\/ (4 + 7*cos(x))
$$- \frac{\frac{7 \left(4 \cos{\left(x \right)} + 7\right) \sin{\left(x \right)}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} - \frac{4 \sin{\left(x \right)}}{7 \cos{\left(x \right)} + 4}}{\sqrt{- \frac{\left(4 \cos{\left(x \right)} + 7\right)^{2}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} + 1}}$$
The second derivative
[src]
/ 2 \
| / 7*(7 + 4*cos(x))\ 2 |
| 2 2 |-4 + ----------------| *sin (x)*(7 + 4*cos(x))|
| 56*sin (x) 7*(7 + 4*cos(x))*cos(x) 98*sin (x)*(7 + 4*cos(x)) \ 4 + 7*cos(x) / |
-|-4*cos(x) - ------------ + ----------------------- + ------------------------- + -----------------------------------------------|
| 4 + 7*cos(x) 4 + 7*cos(x) 2 / 2\ |
| (4 + 7*cos(x)) | (7 + 4*cos(x)) | 2 |
| |1 - ---------------|*(4 + 7*cos(x)) |
| | 2| |
\ \ (4 + 7*cos(x)) / /
------------------------------------------------------------------------------------------------------------------------------------
_____________________
/ 2
/ (7 + 4*cos(x))
/ 1 - --------------- *(4 + 7*cos(x))
/ 2
\/ (4 + 7*cos(x))
$$- \frac{\frac{\left(\frac{7 \left(4 \cos{\left(x \right)} + 7\right)}{7 \cos{\left(x \right)} + 4} - 4\right)^{2} \left(4 \cos{\left(x \right)} + 7\right) \sin^{2}{\left(x \right)}}{\left(- \frac{\left(4 \cos{\left(x \right)} + 7\right)^{2}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} + 1\right) \left(7 \cos{\left(x \right)} + 4\right)^{2}} + \frac{7 \left(4 \cos{\left(x \right)} + 7\right) \cos{\left(x \right)}}{7 \cos{\left(x \right)} + 4} + \frac{98 \left(4 \cos{\left(x \right)} + 7\right) \sin^{2}{\left(x \right)}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} - 4 \cos{\left(x \right)} - \frac{56 \sin^{2}{\left(x \right)}}{7 \cos{\left(x \right)} + 4}}{\sqrt{- \frac{\left(4 \cos{\left(x \right)} + 7\right)^{2}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} + 1} \left(7 \cos{\left(x \right)} + 4\right)}$$
The third derivative
[src]
/ / 2 2 2 2 \ / 2 2 \\
| / 7*(7 + 4*cos(x))\ | 2 112*sin (x)*(7 + 4*cos(x)) 7*(7 + 4*cos(x)) *cos(x) 147*(7 + 4*cos(x)) *sin (x)| 3 / 7*(7 + 4*cos(x))\ | 56*sin (x) 7*(7 + 4*cos(x))*cos(x) 98*sin (x)*(7 + 4*cos(x))||
| |-4 + ----------------|*|16*sin (x) - 4*(7 + 4*cos(x))*cos(x) - -------------------------- + ------------------------ + ---------------------------| / 7*(7 + 4*cos(x))\ 2 2 2*|-4 + ----------------|*(7 + 4*cos(x))*|-4*cos(x) - ------------ + ----------------------- + -------------------------||
| 2 2 \ 4 + 7*cos(x) / | 4 + 7*cos(x) 4 + 7*cos(x) 2 | 3*|-4 + ----------------| *(7 + 4*cos(x)) *sin (x) \ 4 + 7*cos(x) / | 4 + 7*cos(x) 4 + 7*cos(x) 2 ||
| 7*(7 + 4*cos(x)) 168*cos(x) 1176*sin (x) 2058*sin (x)*(7 + 4*cos(x)) 294*(7 + 4*cos(x))*cos(x) \ (4 + 7*cos(x)) / \ 4 + 7*cos(x) / \ (4 + 7*cos(x)) /|
|-4 + ---------------- + ------------ + --------------- - --------------------------- - ------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------|*sin(x)
| 4 + 7*cos(x) 4 + 7*cos(x) 2 3 2 / 2\ 2 / 2\ |
| (4 + 7*cos(x)) (4 + 7*cos(x)) (4 + 7*cos(x)) | (7 + 4*cos(x)) | 2 / 2\ | (7 + 4*cos(x)) | 2 |
| |1 - ---------------|*(4 + 7*cos(x)) | (7 + 4*cos(x)) | 4 |1 - ---------------|*(4 + 7*cos(x)) |
| | 2| |1 - ---------------| *(4 + 7*cos(x)) | 2| |
| \ (4 + 7*cos(x)) / | 2| \ (4 + 7*cos(x)) / |
\ \ (4 + 7*cos(x)) / /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
_____________________
/ 2
/ (7 + 4*cos(x))
/ 1 - --------------- *(4 + 7*cos(x))
/ 2
\/ (4 + 7*cos(x))
$$\frac{\left(- \frac{3 \left(\frac{7 \left(4 \cos{\left(x \right)} + 7\right)}{7 \cos{\left(x \right)} + 4} - 4\right)^{3} \left(4 \cos{\left(x \right)} + 7\right)^{2} \sin^{2}{\left(x \right)}}{\left(- \frac{\left(4 \cos{\left(x \right)} + 7\right)^{2}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} + 1\right)^{2} \left(7 \cos{\left(x \right)} + 4\right)^{4}} - \frac{2 \left(\frac{7 \left(4 \cos{\left(x \right)} + 7\right)}{7 \cos{\left(x \right)} + 4} - 4\right) \left(4 \cos{\left(x \right)} + 7\right) \left(\frac{7 \left(4 \cos{\left(x \right)} + 7\right) \cos{\left(x \right)}}{7 \cos{\left(x \right)} + 4} + \frac{98 \left(4 \cos{\left(x \right)} + 7\right) \sin^{2}{\left(x \right)}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} - 4 \cos{\left(x \right)} - \frac{56 \sin^{2}{\left(x \right)}}{7 \cos{\left(x \right)} + 4}\right)}{\left(- \frac{\left(4 \cos{\left(x \right)} + 7\right)^{2}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} + 1\right) \left(7 \cos{\left(x \right)} + 4\right)^{2}} - \frac{\left(\frac{7 \left(4 \cos{\left(x \right)} + 7\right)}{7 \cos{\left(x \right)} + 4} - 4\right) \left(\frac{7 \left(4 \cos{\left(x \right)} + 7\right)^{2} \cos{\left(x \right)}}{7 \cos{\left(x \right)} + 4} + \frac{147 \left(4 \cos{\left(x \right)} + 7\right)^{2} \sin^{2}{\left(x \right)}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} - 4 \left(4 \cos{\left(x \right)} + 7\right) \cos{\left(x \right)} - \frac{112 \left(4 \cos{\left(x \right)} + 7\right) \sin^{2}{\left(x \right)}}{7 \cos{\left(x \right)} + 4} + 16 \sin^{2}{\left(x \right)}\right)}{\left(- \frac{\left(4 \cos{\left(x \right)} + 7\right)^{2}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} + 1\right) \left(7 \cos{\left(x \right)} + 4\right)^{2}} + \frac{7 \left(4 \cos{\left(x \right)} + 7\right)}{7 \cos{\left(x \right)} + 4} - \frac{294 \left(4 \cos{\left(x \right)} + 7\right) \cos{\left(x \right)}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} - \frac{2058 \left(4 \cos{\left(x \right)} + 7\right) \sin^{2}{\left(x \right)}}{\left(7 \cos{\left(x \right)} + 4\right)^{3}} - 4 + \frac{168 \cos{\left(x \right)}}{7 \cos{\left(x \right)} + 4} + \frac{1176 \sin^{2}{\left(x \right)}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}}\right) \sin{\left(x \right)}}{\sqrt{- \frac{\left(4 \cos{\left(x \right)} + 7\right)^{2}}{\left(7 \cos{\left(x \right)} + 4\right)^{2}} + 1} \left(7 \cos{\left(x \right)} + 4\right)}$$