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How to use it?
- Derivative of:
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Derivative of x^2*sqrt(1-x^2)
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Derivative of e^x*x^2
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Derivative of (x+3)/(x-3)
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Derivative of x^3/(x^2-4)
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Identical expressions
- one /((x- two)*(x^ two - one))
- 1 divide by ((x minus 2) multiply by (x squared minus 1))
- one divide by ((x minus two) multiply by (x to the power of two minus one))
- 1/((x-2)*(x2-1))
- 1/x-2*x2-1
- 1/((x-2)*(x²-1))
- 1/((x-2)*(x to the power of 2-1))
- 1/((x-2)(x^2-1))
- 1/((x-2)(x2-1))
- 1/x-2x2-1
- 1/x-2x^2-1
- 1 divide by ((x-2)*(x^2-1))
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Similar expressions
- 1/((x-2)*(x^2+1))
- 1/((x+2)*(x^2-1))
Derivative of 1/((x-2)*(x^2-1))
The solution
You have entered
[src]
1
1*----------------
/ 2 \
(x - 2)*\x - 1/
$$1 \cdot \frac{1}{\left(x - 2\right) \left(x^{2} - 1\right)}$$
d / 1 \ --|1*----------------| dx| / 2 \| \ (x - 2)*\x - 1//
$$\frac{d}{d x} 1 \cdot \frac{1}{\left(x - 2\right) \left(x^{2} - 1\right)}$$
Detail solution
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Apply the quotient rule, which is:
and .
To find :
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The derivative of the constant is zero.
To find :
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Apply the product rule:
; 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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Apply the power rule: goes to
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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Apply the power rule: goes to
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 answer is:
The first derivative
[src]
1 / 2 \
----------------*\1 - x - 2*x*(x - 2)/
/ 2 \
(x - 2)*\x - 1/
---------------------------------------
/ 2 \
(x - 2)*\x - 1/
$$\frac{\frac{1}{\left(x - 2\right) \left(x^{2} - 1\right)} \left(- x^{2} - 2 x \left(x - 2\right) + 1\right)}{\left(x - 2\right) \left(x^{2} - 1\right)}$$
The second derivative
[src]
2 / 2 \
-1 + x + 2*x*(-2 + x) / 1 2*x \ / 2 \ 2*x*\-1 + x + 2*x*(-2 + x)/
4 - 6*x + ---------------------- + |------ + -------|*\-1 + x + 2*x*(-2 + x)/ + ----------------------------
-2 + x |-2 + x 2| 2
\ -1 + x / -1 + x
-------------------------------------------------------------------------------------------------------------
2
/ 2\ 2
\-1 + x / *(-2 + x)
$$\frac{- 6 x + \frac{2 x \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{x^{2} - 1} + \left(\frac{2 x}{x^{2} - 1} + \frac{1}{x - 2}\right) \left(x^{2} + 2 x \left(x - 2\right) - 1\right) + 4 + \frac{x^{2} + 2 x \left(x - 2\right) - 1}{x - 2}}{\left(x - 2\right)^{2} \left(x^{2} - 1\right)^{2}}$$
The third derivative
[src]
/ / 1 2*x \ / 2 \ / 1 2*x \ / 2 \ \
| |------ + -------|*\-1 + x + 2*x*(-2 + x)/ 2*x*|------ + -------|*\-1 + x + 2*x*(-2 + x)/ |
| / 2 \ / 2 \ / 2 \ |-2 + x 2| 2 / 2 \ |-2 + x 2| / 2 \|
| 6*(-2 + 3*x) 2*\-1 + x + 2*x*(-2 + x)/ / 1 2*x \ / 2 \ | 1 1 4*x 2*x | 3*\-1 + x + 2*x*(-2 + x)/ \ -1 + x / 12*x*(-2 + 3*x) 12*x *\-1 + x + 2*x*(-2 + x)/ \ -1 + x / 8*x*\-1 + x + 2*x*(-2 + x)/|
-|6 - ------------ - -------------------------- - 2*(-2 + 3*x)*|------ + -------| + 2*\-1 + x + 2*x*(-2 + x)/*|--------- - ------- + ---------- + ------------------| + -------------------------- + ------------------------------------------- - --------------- + ------------------------------ + ----------------------------------------------- + ----------------------------|
| -2 + x 2 |-2 + x 2| | 2 2 2 / 2\ | 2 -2 + x 2 2 2 / 2\ |
| -1 + x \ -1 + x / |(-2 + x) -1 + x / 2\ \-1 + x /*(-2 + x)| (-2 + x) -1 + x / 2\ -1 + x \-1 + x /*(-2 + x) |
\ \ \-1 + x / / \-1 + x / /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
/ 2\ 2
\-1 + x / *(-2 + x)
$$- \frac{\frac{12 x^{2} \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{\left(x^{2} - 1\right)^{2}} - \frac{12 x \left(3 x - 2\right)}{x^{2} - 1} + \frac{2 x \left(\frac{2 x}{x^{2} - 1} + \frac{1}{x - 2}\right) \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{x^{2} - 1} + \frac{8 x \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{\left(x - 2\right) \left(x^{2} - 1\right)} - 2 \cdot \left(3 x - 2\right) \left(\frac{2 x}{x^{2} - 1} + \frac{1}{x - 2}\right) + 2 \left(x^{2} + 2 x \left(x - 2\right) - 1\right) \left(\frac{4 x^{2}}{\left(x^{2} - 1\right)^{2}} + \frac{2 x}{\left(x - 2\right) \left(x^{2} - 1\right)} - \frac{1}{x^{2} - 1} + \frac{1}{\left(x - 2\right)^{2}}\right) + 6 - \frac{2 \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{x^{2} - 1} - \frac{6 \cdot \left(3 x - 2\right)}{x - 2} + \frac{\left(\frac{2 x}{x^{2} - 1} + \frac{1}{x - 2}\right) \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{x - 2} + \frac{3 \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{\left(x - 2\right)^{2}}}{\left(x - 2\right)^{2} \left(x^{2} - 1\right)^{2}}$$
3-я производная
[src]
/ / 1 2*x \ / 2 \ / 1 2*x \ / 2 \ \
| |------ + -------|*\-1 + x + 2*x*(-2 + x)/ 2*x*|------ + -------|*\-1 + x + 2*x*(-2 + x)/ |
| / 2 \ / 2 \ / 2 \ |-2 + x 2| 2 / 2 \ |-2 + x 2| / 2 \|
| 6*(-2 + 3*x) 2*\-1 + x + 2*x*(-2 + x)/ / 1 2*x \ / 2 \ | 1 1 4*x 2*x | 3*\-1 + x + 2*x*(-2 + x)/ \ -1 + x / 12*x*(-2 + 3*x) 12*x *\-1 + x + 2*x*(-2 + x)/ \ -1 + x / 8*x*\-1 + x + 2*x*(-2 + x)/|
-|6 - ------------ - -------------------------- - 2*(-2 + 3*x)*|------ + -------| + 2*\-1 + x + 2*x*(-2 + x)/*|--------- - ------- + ---------- + ------------------| + -------------------------- + ------------------------------------------- - --------------- + ------------------------------ + ----------------------------------------------- + ----------------------------|
| -2 + x 2 |-2 + x 2| | 2 2 2 / 2\ | 2 -2 + x 2 2 2 / 2\ |
| -1 + x \ -1 + x / |(-2 + x) -1 + x / 2\ \-1 + x /*(-2 + x)| (-2 + x) -1 + x / 2\ -1 + x \-1 + x /*(-2 + x) |
\ \ \-1 + x / / \-1 + x / /
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2
/ 2\ 2
\-1 + x / *(-2 + x)
$$- \frac{\frac{12 x^{2} \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{\left(x^{2} - 1\right)^{2}} - \frac{12 x \left(3 x - 2\right)}{x^{2} - 1} + \frac{2 x \left(\frac{2 x}{x^{2} - 1} + \frac{1}{x - 2}\right) \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{x^{2} - 1} + \frac{8 x \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{\left(x - 2\right) \left(x^{2} - 1\right)} - 2 \cdot \left(3 x - 2\right) \left(\frac{2 x}{x^{2} - 1} + \frac{1}{x - 2}\right) + 2 \left(x^{2} + 2 x \left(x - 2\right) - 1\right) \left(\frac{4 x^{2}}{\left(x^{2} - 1\right)^{2}} + \frac{2 x}{\left(x - 2\right) \left(x^{2} - 1\right)} - \frac{1}{x^{2} - 1} + \frac{1}{\left(x - 2\right)^{2}}\right) + 6 - \frac{2 \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{x^{2} - 1} - \frac{6 \cdot \left(3 x - 2\right)}{x - 2} + \frac{\left(\frac{2 x}{x^{2} - 1} + \frac{1}{x - 2}\right) \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{x - 2} + \frac{3 \left(x^{2} + 2 x \left(x - 2\right) - 1\right)}{\left(x - 2\right)^{2}}}{\left(x - 2\right)^{2} \left(x^{2} - 1\right)^{2}}$$
The graph