Mister Exam
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- 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 3^x
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Derivative of x^(1/2)
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Derivative of x*atan(x)
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Derivative of x*acot(x)
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Identical expressions
- sqrt(x)*sin(x)/(one -cos(x))
- square root of (x) multiply by sinus of (x) divide by (1 minus co sinus of e of (x))
- square root of (x) multiply by sinus of (x) divide by (one minus co sinus of e of (x))
- √(x)*sin(x)/(1-cos(x))
- sqrt(x)sin(x)/(1-cos(x))
- sqrtxsinx/1-cosx
- sqrt(x)*sin(x) divide by (1-cos(x))
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Similar expressions
- sqrt(x)*sin(x)/(1+cos(x))
- sqrt(x)*sinx/(1-cosx)
Derivative of sqrt(x)*sin(x)/(1-cos(x))
The solution
You have entered
[src]
___ \/ x *sin(x) ------------ 1 - cos(x)
$$\frac{\sqrt{x} \sin{\left(x \right)}}{1 - \cos{\left(x \right)}}$$
/ ___ \ d |\/ x *sin(x)| --|------------| dx\ 1 - cos(x) /
$$\frac{d}{d x} \frac{\sqrt{x} \sin{\left(x \right)}}{1 - \cos{\left(x \right)}}$$
Detail solution
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Apply the quotient rule, which is:
and .
To find :
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Apply the product rule:
; to find :
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Apply the power rule: goes to
; to find :
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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 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
\/ x *cos(x) sin(x) \/ x *sin (x)
------------ + -------------------- - -------------
1 - cos(x) ___ 2
2*\/ x *(1 - cos(x)) (1 - cos(x))
$$\frac{\sqrt{x} \cos{\left(x \right)}}{1 - \cos{\left(x \right)}} - \frac{\sqrt{x} \sin^{2}{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} + \frac{\sin{\left(x \right)}}{2 \sqrt{x} \left(1 - \cos{\left(x \right)}\right)}$$
The second derivative
[src]
/ 2 \
___ | 2*sin (x) |
2 \/ x *|----------- + cos(x)|*sin(x) ___
___ cos(x) sin(x) sin (x) \-1 + cos(x) / 2*\/ x *cos(x)*sin(x)
\/ x *sin(x) - ------ + ------ - ------------------- - ----------------------------------- - ---------------------
___ 3/2 ___ -1 + cos(x) -1 + cos(x)
\/ x 4*x \/ x *(-1 + cos(x))
------------------------------------------------------------------------------------------------------------------
-1 + cos(x)
$$\frac{\sqrt{x} \sin{\left(x \right)} - \frac{\sqrt{x} \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \sqrt{x} \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{\cos{\left(x \right)}}{\sqrt{x}} - \frac{\sin^{2}{\left(x \right)}}{\sqrt{x} \left(\cos{\left(x \right)} - 1\right)} + \frac{\sin{\left(x \right)}}{4 x^{\frac{3}{2}}}}{\cos{\left(x \right)} - 1}$$
The third derivative
[src]
/ 2 \
___ 2 | 6*cos(x) 6*sin (x) | / 2 \ / 2 \
\/ x *sin (x)*|-1 + ----------- + --------------| ___ | 2*sin (x) | | 2*sin (x) |
___ 2 2 | -1 + cos(x) 2| 3*\/ x *|----------- + cos(x)|*cos(x) 3*|----------- + cos(x)|*sin(x)
___ 3*sin(x) 3*sin(x) 3*cos(x) 3*\/ x *sin (x) 3*sin (x) \ (-1 + cos(x)) / \-1 + cos(x) / 3*cos(x)*sin(x) \-1 + cos(x) /
\/ x *cos(x) - -------- + -------- + -------- + --------------- + -------------------- - ------------------------------------------------- - ------------------------------------- - ------------------- - -------------------------------
5/2 ___ 3/2 -1 + cos(x) 3/2 -1 + cos(x) -1 + cos(x) ___ ___
8*x 2*\/ x 4*x 4*x *(-1 + cos(x)) \/ x *(-1 + cos(x)) 2*\/ x *(-1 + cos(x))
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
-1 + cos(x)
$$\frac{\sqrt{x} \cos{\left(x \right)} - \frac{3 \sqrt{x} \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{\sqrt{x} \left(-1 + \frac{6 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{6 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}}\right) \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{3 \sqrt{x} \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{3 \sin{\left(x \right)}}{2 \sqrt{x}} - \frac{3 \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \sqrt{x} \left(\cos{\left(x \right)} - 1\right)} - \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{x} \left(\cos{\left(x \right)} - 1\right)} + \frac{3 \cos{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \sin^{2}{\left(x \right)}}{4 x^{\frac{3}{2}} \left(\cos{\left(x \right)} - 1\right)} - \frac{3 \sin{\left(x \right)}}{8 x^{\frac{5}{2}}}}{\cos{\left(x \right)} - 1}$$
The graph