Mister Exam
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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 (2*x-1)^2
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Derivative of 1/(x+2)
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Derivative of (2*e)^cos(log(6*x-1))^(2)
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Derivative of 1/tg(x)
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Identical expressions
- sqrt((one +sinx)/(one -sinx))
- square root of ((1 plus sinus of x) divide by (1 minus sinus of x))
- square root of ((one plus sinus of x) divide by (one minus sinus of x))
- √((1+sinx)/(1-sinx))
- sqrt1+sinx/1-sinx
- sqrt((1+sinx) divide by (1-sinx))
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Similar expressions
- log(sqrt((1+sin(x))/(1-sin(x))))
- sqrt((1+sinx)/(1+sinx))
- sqrt((1-sinx)/(1-sinx))
Derivative of sqrt((1+sinx)/(1-sinx))
The solution
You have entered
[src]
____________ / 1 + sin(x) / ---------- \/ 1 - sin(x)
$$\sqrt{\frac{\sin{\left(x \right)} + 1}{1 - \sin{\left(x \right)}}}$$
sqrt((1 + sin(x))/(1 - sin(x)))
Detail solution
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Let .
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Apply the power rule: goes to
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Then, apply the chain rule. Multiply by :
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Apply the quotient rule, which is:
and .
To find :
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Differentiate term by term:
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The derivative of the constant is zero.
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The derivative of sine is cosine:
The result is:
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To find :
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Differentiate term by term:
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The derivative of the constant is zero.
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The derivative of a constant times a function is the constant times the derivative of the function.
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The derivative of sine is cosine:
So, the result is:
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The result is:
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Now plug in to the quotient rule:
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The result of the chain rule is:
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Now simplify:
The answer is:
The first derivative
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____________
/ 1 + sin(x) / cos(x) (1 + sin(x))*cos(x)\
/ ---------- *(1 - sin(x))*|-------------- + -------------------|
\/ 1 - sin(x) |2*(1 - sin(x)) 2 |
\ 2*(1 - sin(x)) /
--------------------------------------------------------------------
1 + sin(x)
$$\frac{\sqrt{\frac{\sin{\left(x \right)} + 1}{1 - \sin{\left(x \right)}}} \left(1 - \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{2 \left(1 - \sin{\left(x \right)}\right)} + \frac{\left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}{2 \left(1 - \sin{\left(x \right)}\right)^{2}}\right)}{\sin{\left(x \right)} + 1}$$
The second derivative
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/ 2 \
| 2 / 1 + sin(x)\ 2 / 1 + sin(x)\ / 1 + sin(x)\ 2 |
________________ | 2 2 cos (x)*|1 - -----------| cos (x)*|1 - -----------| |1 - -----------| *cos (x)|
/ -(1 + sin(x)) | sin(x) cos (x) cos (x)*(1 + sin(x)) \ -1 + sin(x)/ (1 + sin(x))*sin(x) \ -1 + sin(x)/ \ -1 + sin(x)/ |
/ -------------- *|- ------ - ----------- + -------------------- + ------------------------- + ------------------- - ------------------------- + --------------------------|
\/ -1 + sin(x) | 2 -1 + sin(x) 2 2*(-1 + sin(x)) 2*(-1 + sin(x)) 2*(1 + sin(x)) 4*(1 + sin(x)) |
\ (-1 + sin(x)) /
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1 + sin(x)
$$\frac{\sqrt{- \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}} \left(\frac{\left(1 - \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}\right)^{2} \cos^{2}{\left(x \right)}}{4 \left(\sin{\left(x \right)} + 1\right)} - \frac{\left(1 - \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}\right) \cos^{2}{\left(x \right)}}{2 \left(\sin{\left(x \right)} + 1\right)} + \frac{\left(1 - \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}\right) \cos^{2}{\left(x \right)}}{2 \left(\sin{\left(x \right)} - 1\right)} - \frac{\sin{\left(x \right)}}{2} + \frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{2 \left(\sin{\left(x \right)} - 1\right)} - \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{\left(\sin{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}}\right)}{\sin{\left(x \right)} + 1}$$
The third derivative
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/ 2 2 2 2 / 2 2 \ \
| 2*cos (x) (1 + sin(x))*sin(x) 2*cos (x)*(1 + sin(x)) 2*cos (x) (1 + sin(x))*sin(x) 2*cos (x)*(1 + sin(x)) 2 / 1 + sin(x)\ | 2*cos (x) (1 + sin(x))*sin(x) 2*cos (x)*(1 + sin(x)) | 3 2 |
| ----------- - ------------------- - ---------------------- + sin(x) ----------- - ------------------- - ---------------------- + sin(x) 2 / 1 + sin(x)\ / 1 + sin(x)\ / 1 + sin(x)\ 2 3*|1 - -----------|*|----------- - ------------------- - ---------------------- + sin(x)| / 1 + sin(x)\ / 1 + sin(x)\ 2 2 / 1 + sin(x)\ / 1 + sin(x)\ 2 |
________________ | -1 + sin(x) -1 + sin(x) 2 -1 + sin(x) -1 + sin(x) 2 2 cos (x)*|1 - -----------| |1 - -----------|*sin(x) 2 3*|1 - -----------| *cos (x) \ -1 + sin(x)/ |-1 + sin(x) -1 + sin(x) 2 | |1 - -----------|*sin(x) |1 - -----------| *cos (x) cos (x)*|1 - -----------| 3*|1 - -----------| *cos (x)|
/ -(1 + sin(x)) | 1 (-1 + sin(x)) 1 + sin(x) (-1 + sin(x)) 3*sin(x) 3*cos (x) \ -1 + sin(x)/ \ -1 + sin(x)/ 3*cos (x)*(1 + sin(x)) 3*(1 + sin(x))*sin(x) \ -1 + sin(x)/ \ (-1 + sin(x)) / \ -1 + sin(x)/ \ -1 + sin(x)/ \ -1 + sin(x)/ \ -1 + sin(x)/ |
/ -------------- *|- - + ------------------------------------------------------------------- + --------------- - ------------------------------------------------------------------- + ----------- + -------------- + ------------------------- + ------------------------ - ---------------------- - --------------------- - ---------------------------- - ----------------------------------------------------------------------------------------- - ------------------------ + -------------------------- - -------------------------- + ----------------------------|*cos(x)
\/ -1 + sin(x) | 2 1 + sin(x) 2*(-1 + sin(x)) -1 + sin(x) -1 + sin(x) 2 2 2*(1 + sin(x)) 3 2 2 4*(1 + sin(x)) 2*(-1 + sin(x)) 2 (1 + sin(x))*(-1 + sin(x)) 4*(1 + sin(x))*(-1 + sin(x))|
\ (-1 + sin(x)) (1 + sin(x)) (-1 + sin(x)) (-1 + sin(x)) 4*(1 + sin(x)) 8*(1 + sin(x)) /
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1 + sin(x)
$$\frac{\sqrt{- \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}} \left(\frac{\left(1 - \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}\right)^{3} \cos^{2}{\left(x \right)}}{8 \left(\sin{\left(x \right)} + 1\right)^{2}} - \frac{3 \left(1 - \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}\right)^{2} \cos^{2}{\left(x \right)}}{4 \left(\sin{\left(x \right)} + 1\right)^{2}} + \frac{3 \left(1 - \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}\right)^{2} \cos^{2}{\left(x \right)}}{4 \left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 1\right)} - \frac{3 \left(1 - \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}\right) \left(\sin{\left(x \right)} - \frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1} - \frac{2 \left(\sin{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}}\right)}{4 \left(\sin{\left(x \right)} + 1\right)} + \frac{\left(1 - \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \left(\sin{\left(x \right)} + 1\right)} + \frac{\left(1 - \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}\right) \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} - \frac{\left(1 - \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \left(\sin{\left(x \right)} - 1\right)} - \frac{\left(1 - \frac{\sin{\left(x \right)} + 1}{\sin{\left(x \right)} - 1}\right) \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 1\right)} - \frac{1}{2} + \frac{\sin{\left(x \right)} - \frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1} - \frac{2 \left(\sin{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}}}{\sin{\left(x \right)} + 1} + \frac{\sin{\left(x \right)} + 1}{2 \left(\sin{\left(x \right)} - 1\right)} - \frac{\sin{\left(x \right)} - \frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1} - \frac{2 \left(\sin{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}}}{\sin{\left(x \right)} - 1} + \frac{3 \sin{\left(x \right)}}{\sin{\left(x \right)} - 1} - \frac{3 \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}} + \frac{3 \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}} - \frac{3 \left(\sin{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{3}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)} + 1}$$
The graph