Mister Exam
Other calculators
- Derivative of a implicit defined function
- Derivative of Parametric Function
- Partial derivative of the function
- Curve tracing functions Step by Step
- Integral Step by Step
- Differential equations Step by Step
- Limits Step by Step
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How to use it?
- Derivative of:
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Derivative of f(x)=2x-3
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Derivative of 2*cos(x)+sin(x)
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Derivative of (1-x^2)^10
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Derivative of (-1)/x+4*x^2
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Identical expressions
- sin(t)/(one -cos(t))
- sinus of (t) divide by (1 minus co sinus of e of (t))
- sinus of (t) divide by (one minus co sinus of e of (t))
- sint/1-cost
- sin(t) divide by (1-cos(t))
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Similar expressions
- sint/(1-cost)
- sint/1-cost
- sin(t)/(1+cos(t))
- y=sint/(1-cost)
Derivative of sin(t)/(1-cos(t))
The solution
You have entered
[src]
sin(t) ---------- 1 - cos(t)
$$\frac{\sin{\left(t \right)}}{1 - \cos{\left(t \right)}}$$
d / sin(t) \ --|----------| dt\1 - cos(t)/
$$\frac{d}{d t} \frac{\sin{\left(t \right)}}{1 - \cos{\left(t \right)}}$$
Detail solution
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Apply the quotient rule, which is:
and .
To find :
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The derivative of sine is cosine:
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 cosine is negative sine:
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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Now simplify:
The answer is:
The first derivative
[src]
2
cos(t) sin (t)
---------- - -------------
1 - cos(t) 2
(1 - cos(t))
$$\frac{\cos{\left(t \right)}}{1 - \cos{\left(t \right)}} - \frac{\sin^{2}{\left(t \right)}}{\left(1 - \cos{\left(t \right)}\right)^{2}}$$
The second derivative
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/ 2 \
| 2*sin (t) |
| ----------- + cos(t) |
| -1 + cos(t) 2*cos(t) |
|1 - -------------------- - -----------|*sin(t)
\ -1 + cos(t) -1 + cos(t)/
-----------------------------------------------
-1 + cos(t)
$$\frac{\left(1 - \frac{\cos{\left(t \right)} + \frac{2 \sin^{2}{\left(t \right)}}{\cos{\left(t \right)} - 1}}{\cos{\left(t \right)} - 1} - \frac{2 \cos{\left(t \right)}}{\cos{\left(t \right)} - 1}\right) \sin{\left(t \right)}}{\cos{\left(t \right)} - 1}$$
The third derivative
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/ 2 \
2 | 6*cos(t) 6*sin (t) | / 2 \
sin (t)*|-1 + ----------- + --------------| | 2*sin (t) |
2 | -1 + cos(t) 2| 3*|----------- + cos(t)|*cos(t)
3*sin (t) \ (-1 + cos(t)) / \-1 + cos(t) /
----------- - ------------------------------------------- - ------------------------------- + cos(t)
-1 + cos(t) -1 + cos(t) -1 + cos(t)
----------------------------------------------------------------------------------------------------
-1 + cos(t)
$$\frac{\cos{\left(t \right)} - \frac{3 \left(\cos{\left(t \right)} + \frac{2 \sin^{2}{\left(t \right)}}{\cos{\left(t \right)} - 1}\right) \cos{\left(t \right)}}{\cos{\left(t \right)} - 1} - \frac{\left(-1 + \frac{6 \cos{\left(t \right)}}{\cos{\left(t \right)} - 1} + \frac{6 \sin^{2}{\left(t \right)}}{\left(\cos{\left(t \right)} - 1\right)^{2}}\right) \sin^{2}{\left(t \right)}}{\cos{\left(t \right)} - 1} + \frac{3 \sin^{2}{\left(t \right)}}{\cos{\left(t \right)} - 1}}{\cos{\left(t \right)} - 1}$$
The graph