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How to use it?
- Derivative of:
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Derivative of x^3/cosx
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Derivative of 3^-x
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Derivative of (x-6)x^3
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Derivative of (x-5)^2*e^x-7
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Identical expressions
- y=sqrt(x*(x- one)²)
- y equally square root of (x multiply by (x minus 1)²)
- y equally square root of (x multiply by (x minus one)²)
- y=√(x*(x-1)²)
- y=sqrt(x(x-1)²)
- y=sqrtxx-1²
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Similar expressions
- y=sqrt(x*(x+1)²)
Derivative of y=sqrt(x*(x-1)²)
The solution
You have entered
[src]
____________ / 2 \/ x*(x - 1)
$$\sqrt{x \left(x - 1\right)^{2}}$$
sqrt(x*(x - 1)^2)
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 product rule:
; to find :
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Apply the power rule: goes to
; to find :
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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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Differentiate term by term:
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Apply the power rule: goes to
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The derivative of the constant is zero.
The result is:
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The result of the chain rule is:
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The result is:
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The result of the chain rule is:
Now simplify:
The answer is:
The first derivative
[src]
/ 2 \
___ |(x - 1) x*(-2 + 2*x)|
\/ x *|x - 1|*|-------- + ------------|
\ 2 2 /
---------------------------------------
2
x*(x - 1)
$$\frac{\sqrt{x} \left|{x - 1}\right| \left(\frac{x \left(2 x - 2\right)}{2} + \frac{\left(x - 1\right)^{2}}{2}\right)}{x \left(x - 1\right)^{2}}$$
The second derivative
[src]
/|-1 + x| ___ \
(-1 + x)*(-1 + 3*x)*|-------- + 2*\/ x *sign(-1 + x)|
| ___ |
(-2 + 3*x)*|-1 + x| (-1 + 3*x)*|-1 + x| (-1 + x)*(-1 + 3*x)*|-1 + x| \ \/ x /
------------------- - ------------------- - ---------------------------- + -----------------------------------------------------
___ ___ 3/2 4*x
\/ x \/ x 2*x
--------------------------------------------------------------------------------------------------------------------------------
2
(-1 + x)
$$\frac{\frac{\left(x - 1\right) \left(3 x - 1\right) \left(2 \sqrt{x} \operatorname{sign}{\left(x - 1 \right)} + \frac{\left|{x - 1}\right|}{\sqrt{x}}\right)}{4 x} + \frac{\left(3 x - 2\right) \left|{x - 1}\right|}{\sqrt{x}} - \frac{\left(3 x - 1\right) \left|{x - 1}\right|}{\sqrt{x}} - \frac{\left(x - 1\right) \left(3 x - 1\right) \left|{x - 1}\right|}{2 x^{\frac{3}{2}}}}{\left(x - 1\right)^{2}}$$
The third derivative
[src]
/|-1 + x| ___ \ /|-1 + x| ___ \ /|-1 + x| ___ \ / |-1 + x| 4*sign(-1 + x) ___ \
(-2 + 3*x)*|-------- + 2*\/ x *sign(-1 + x)| (-1 + 3*x)*|-------- + 2*\/ x *sign(-1 + x)| (-1 + x)*(-1 + 3*x)*|-------- + 2*\/ x *sign(-1 + x)| (-1 + x)*(-1 + 3*x)*|- -------- + -------------- + 8*\/ x *DiracDelta(-1 + x)|
| ___ | | ___ | | ___ | | 3/2 ___ |
3*|-1 + x| (-2 + 3*x)*sign(-1 + x) \ \/ x / (-1 + 3*x)*sign(-1 + x) 3*(-2 + 3*x)*|-1 + x| \ \/ x / 3*(-1 + 3*x)*|-1 + x| 4*(-2 + 3*x)*|-1 + x| 3*(-1 + 3*x)*|-1 + x| (-1 + x)*(-1 + 3*x)*sign(-1 + x) \ \/ x / \ x \/ x / 3*(-1 + x)*(-1 + 3*x)*|-1 + x|
---------- + ----------------------- + -------------------------------------------- - ----------------------- - --------------------- - -------------------------------------------- + --------------------- - --------------------- + --------------------- - -------------------------------- - ----------------------------------------------------- + ------------------------------------------------------------------------------ + ------------------------------
___ ___ 2*x ___ 3/2 2*x 3/2 ___ ___ 3/2 2 8*x 5/2
\/ x \/ x \/ x 2*x 2*x \/ x *(-1 + x) \/ x *(-1 + x) 2*x 4*x 4*x
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
(-1 + x)
$$\frac{\frac{\left(x - 1\right) \left(3 x - 1\right) \left(8 \sqrt{x} \delta\left(x - 1\right) + \frac{4 \operatorname{sign}{\left(x - 1 \right)}}{\sqrt{x}} - \frac{\left|{x - 1}\right|}{x^{\frac{3}{2}}}\right)}{8 x} + \frac{\left(3 x - 2\right) \left(2 \sqrt{x} \operatorname{sign}{\left(x - 1 \right)} + \frac{\left|{x - 1}\right|}{\sqrt{x}}\right)}{2 x} - \frac{\left(3 x - 1\right) \left(2 \sqrt{x} \operatorname{sign}{\left(x - 1 \right)} + \frac{\left|{x - 1}\right|}{\sqrt{x}}\right)}{2 x} - \frac{\left(x - 1\right) \left(3 x - 1\right) \left(2 \sqrt{x} \operatorname{sign}{\left(x - 1 \right)} + \frac{\left|{x - 1}\right|}{\sqrt{x}}\right)}{4 x^{2}} + \frac{\left(3 x - 2\right) \operatorname{sign}{\left(x - 1 \right)}}{\sqrt{x}} - \frac{\left(3 x - 1\right) \operatorname{sign}{\left(x - 1 \right)}}{\sqrt{x}} + \frac{3 \left|{x - 1}\right|}{\sqrt{x}} - \frac{4 \left(3 x - 2\right) \left|{x - 1}\right|}{\sqrt{x} \left(x - 1\right)} + \frac{3 \left(3 x - 1\right) \left|{x - 1}\right|}{\sqrt{x} \left(x - 1\right)} - \frac{\left(x - 1\right) \left(3 x - 1\right) \operatorname{sign}{\left(x - 1 \right)}}{2 x^{\frac{3}{2}}} - \frac{3 \left(3 x - 2\right) \left|{x - 1}\right|}{2 x^{\frac{3}{2}}} + \frac{3 \left(3 x - 1\right) \left|{x - 1}\right|}{2 x^{\frac{3}{2}}} + \frac{3 \left(x - 1\right) \left(3 x - 1\right) \left|{x - 1}\right|}{4 x^{\frac{5}{2}}}}{\left(x - 1\right)^{2}}$$