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  • Identical expressions

  • one /(sh(x)*(ch(x)- one))
  • 1 divide by (sh(x) multiply by (ch(x) minus 1))
  • one divide by (sh(x) multiply by (ch(x) minus one))
  • 1/(sh(x)(ch(x)-1))
  • 1/shxchx-1
  • 1 divide by (sh(x)*(ch(x)-1))

  • Similar expressions

  • 1/(sh(x)*(ch(x)+1))
Plotting a function graph/ 1/(sh(x)*(ch(x)-1))

Graphing y = 1/(sh(x)*(ch(x)-1))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                 1          
f(x) = ---------------------
       sinh(x)*(cosh(x) - 1)
f(x)=1(cosh⁡(x)−1)sinh⁡(x)f{\left(x \right)} = \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} 
f = 1/((cosh(x) - 1)*sinh(x))
The graph of the function
0.02.00.20.40.60.81.01.21.41.61.80200000000000
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
1(cosh⁡(x)−1)sinh⁡(x)=0\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 1/(sinh(x)*(cosh(x) - 1)).
1(−1+cosh⁡(0))sinh⁡(0)\frac{1}{\left(-1 + \cosh{\left(0 \right)}\right) \sinh{\left(0 \right)}}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
1(cosh⁡(x)−1)sinh⁡(x)(−(cosh⁡(x)−1)cosh⁡(x)−sinh⁡2(x))(cosh⁡(x)−1)sinh⁡(x)=0\frac{\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} \left(- \left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} - \sinh^{2}{\left(x \right)}\right)}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
((cosh⁡(x)−1)cosh⁡(x)+sinh⁡2(x))(cosh⁡(x)sinh⁡2(x)+1cosh⁡(x)−1)+((cosh⁡(x)−1)cosh⁡(x)+sinh⁡2(x))cosh⁡(x)sinh⁡2(x)+(cosh⁡(x)−1)cosh⁡(x)+sinh⁡2(x)cosh⁡(x)−1−4cosh⁡(x)+1(cosh⁡(x)−1)2sinh⁡(x)=0\frac{\left(\left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} + \sinh^{2}{\left(x \right)}\right) \left(\frac{\cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)}} + \frac{1}{\cosh{\left(x \right)} - 1}\right) + \frac{\left(\left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} + \sinh^{2}{\left(x \right)}\right) \cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)}} + \frac{\left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} + \sinh^{2}{\left(x \right)}}{\cosh{\left(x \right)} - 1} - 4 \cosh{\left(x \right)} + 1}{\left(\cosh{\left(x \right)} - 1\right)^{2} \sinh{\left(x \right)}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞1(cosh⁡(x)−1)sinh⁡(x)=0\lim_{x \to -\infty} \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞1(cosh⁡(x)−1)sinh⁡(x)=0\lim_{x \to \infty} \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 1/(sinh(x)*(cosh(x) - 1)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(1cosh⁡(x)−11sinh⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{\cosh{\left(x \right)} - 1} \frac{1}{\sinh{\left(x \right)}}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(1cosh⁡(x)−11sinh⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{\cosh{\left(x \right)} - 1} \frac{1}{\sinh{\left(x \right)}}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
1(cosh⁡(x)−1)sinh⁡(x)=−1(cosh⁡(x)−1)sinh⁡(x)\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = - \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}}
- No
1(cosh⁡(x)−1)sinh⁡(x)=1(cosh⁡(x)−1)sinh⁡(x)\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}}
- No
so, the function
not is
neither even, nor odd
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