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How to use it?
- Derivative of:
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Derivative of cos(5-3*x)
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Derivative of cos^3(t)
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Derivative of asin(y^2)
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Derivative of (5x^2+2)×(x^2-4)^3
- Graphing y =:
- cbrt((x-1)(x+2)^2)
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Identical expressions
- cbrt((x- one)(x+ two)^ two)
- cubic root of ((x minus 1)(x plus 2) squared )
- cubic root of ((x minus one)(x plus two) to the power of two)
- cbrt((x-1)(x+2)2)
- cbrtx-1x+22
- cbrt((x-1)(x+2)²)
- cbrt((x-1)(x+2) to the power of 2)
- cbrtx-1x+2^2
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Similar expressions
- cbrt((x-1)(x-2)^2)
- cbrt((x+1)(x+2)^2)
Derivative of cbrt((x-1)(x+2)^2)
The solution
You have entered
[src]
__________________ 3 / 2 \/ (x - 1)*(x + 2)
$$\sqrt[3]{\left(x - 1\right) \left(x + 2\right)^{2}}$$
((x - 1)*(x + 2)^2)^(1/3)
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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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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; 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 \
3 _______ 2/3 |(x + 2) (4 + 2*x)*(x - 1)|
\/ x - 1 *|x + 2| *|-------- + -----------------|
\ 3 3 /
---------------------------------------------------
2
(x - 1)*(x + 2)
$$\frac{\sqrt[3]{x - 1} \left|{x + 2}\right|^{\frac{2}{3}} \left(\frac{\left(x - 1\right) \left(2 x + 4\right)}{3} + \frac{\left(x + 2\right)^{2}}{3}\right)}{\left(x - 1\right) \left(x + 2\right)^{2}}$$
The second derivative
[src]
/ 2/3 3 ________ \
| |2 + x| 2*\/ -1 + x *sign(2 + x)|
x*|----------- + ------------------------|
2/3 | 2/3 3 _________ | 2/3 2/3
x*|2 + x| \(-1 + x) \/ |2 + x| / 2*x*|2 + x| 2*|2 + x| *(1 + x)
- ------------ + ------------------------------------------ - ------------------- + --------------------
5/3 3*(-1 + x) 2/3 2/3
(-1 + x) (-1 + x) *(2 + x) (-1 + x) *(2 + x)
--------------------------------------------------------------------------------------------------------
2 + x
$$\frac{\frac{x \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right)} - \frac{2 x \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)} - \frac{x \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{5}{3}}} + \frac{2 \left(x + 1\right) \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)}}{x + 2}$$
3-я производная
[src]
/ 2/3 3 ________ \ / 2/3 3 ________ 2 3 ________ \ / 2/3 3 ________ \ / 2/3 3 ________ \
| |2 + x| 2*\/ -1 + x *sign(2 + x)| | |2 + x| \/ -1 + x *sign (2 + x) 2*sign(2 + x) 6*\/ -1 + x *DiracDelta(2 + x)| | |2 + x| 2*\/ -1 + x *sign(2 + x)| | |2 + x| 2*\/ -1 + x *sign(2 + x)|
x*|----------- + ------------------------| 2*x*|- ----------- - ----------------------- + ----------------------- + ------------------------------| 2*x*|----------- + ------------------------| 2*(1 + x)*|----------- + ------------------------|
2/3 | 2/3 3 _________ | | 5/3 4/3 2/3 3 _________ 3 _________ | 2/3 2/3 2/3 2/3 | 2/3 3 _________ | | 2/3 3 _________ | 2/3
2*|2 + x| \(-1 + x) \/ |2 + x| / \ (-1 + x) |2 + x| (-1 + x) *\/ |2 + x| \/ |2 + x| / 5*x*|2 + x| 8*|2 + x| *(1 + x) 6*x*|2 + x| 10*|2 + x| *(1 + x) \(-1 + x) \/ |2 + x| / 2*x*sign(2 + x) \(-1 + x) \/ |2 + x| / 10*x*|2 + x| 4*x*sign(2 + x) 4*(1 + x)*sign(2 + x)
------------------- - ------------------------------------------ + -------------------------------------------------------------------------------------------------------- + -------------- - -------------------- + -------------------- - --------------------- - -------------------------------------------- - ------------------------- + -------------------------------------------------- + --------------------- - --------------------------------- + ---------------------------------
2/3 2 9*(-1 + x) 8/3 2/3 2 2/3 2 5/3 3*(-1 + x)*(2 + x) 5/3 3 _________ 3*(-1 + x)*(2 + x) 5/3 2/3 3 _________ 2/3 3 _________
(-1 + x) *(2 + x) 3*(-1 + x) 3*(-1 + x) (-1 + x) *(2 + x) (-1 + x) *(2 + x) 3*(-1 + x) *(2 + x) 3*(-1 + x) *\/ |2 + x| 3*(-1 + x) *(2 + x) 3*(-1 + x) *(2 + x)*\/ |2 + x| 3*(-1 + x) *(2 + x)*\/ |2 + x|
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 + x
$$\frac{\frac{2 x \left(\frac{6 \sqrt[3]{x - 1} \delta\left(x + 2\right)}{\sqrt[3]{\left|{x + 2}\right|}} - \frac{\sqrt[3]{x - 1} \operatorname{sign}^{2}{\left(x + 2 \right)}}{\left|{x + 2}\right|^{\frac{4}{3}}} + \frac{2 \operatorname{sign}{\left(x + 2 \right)}}{\left(x - 1\right)^{\frac{2}{3}} \sqrt[3]{\left|{x + 2}\right|}} - \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{5}{3}}}\right)}{9 \left(x - 1\right)} - \frac{2 x \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right) \left(x + 2\right)} - \frac{x \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right)^{2}} - \frac{4 x \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right) \sqrt[3]{\left|{x + 2}\right|}} + \frac{6 x \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)^{2}} - \frac{2 x \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{5}{3}} \sqrt[3]{\left|{x + 2}\right|}} + \frac{10 x \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{5}{3}} \left(x + 2\right)} + \frac{5 x \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{8}{3}}} + \frac{2 \left(x + 1\right) \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right) \left(x + 2\right)} + \frac{4 \left(x + 1\right) \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right) \sqrt[3]{\left|{x + 2}\right|}} - \frac{8 \left(x + 1\right) \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)^{2}} + \frac{2 \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)} - \frac{10 \left(x + 1\right) \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{5}{3}} \left(x + 2\right)}}{x + 2}$$
The third derivative
[src]
/ 2/3 3 ________ \ / 2/3 3 ________ 2 3 ________ \ / 2/3 3 ________ \ / 2/3 3 ________ \
| |2 + x| 2*\/ -1 + x *sign(2 + x)| | |2 + x| \/ -1 + x *sign (2 + x) 2*sign(2 + x) 6*\/ -1 + x *DiracDelta(2 + x)| | |2 + x| 2*\/ -1 + x *sign(2 + x)| | |2 + x| 2*\/ -1 + x *sign(2 + x)|
x*|----------- + ------------------------| 2*x*|- ----------- - ----------------------- + ----------------------- + ------------------------------| 2*x*|----------- + ------------------------| 2*(1 + x)*|----------- + ------------------------|
2/3 | 2/3 3 _________ | | 5/3 4/3 2/3 3 _________ 3 _________ | 2/3 2/3 2/3 2/3 | 2/3 3 _________ | | 2/3 3 _________ | 2/3
2*|2 + x| \(-1 + x) \/ |2 + x| / \ (-1 + x) |2 + x| (-1 + x) *\/ |2 + x| \/ |2 + x| / 5*x*|2 + x| 8*|2 + x| *(1 + x) 6*x*|2 + x| 10*|2 + x| *(1 + x) \(-1 + x) \/ |2 + x| / 2*x*sign(2 + x) \(-1 + x) \/ |2 + x| / 10*x*|2 + x| 4*x*sign(2 + x) 4*(1 + x)*sign(2 + x)
------------------- - ------------------------------------------ + -------------------------------------------------------------------------------------------------------- + -------------- - -------------------- + -------------------- - --------------------- - -------------------------------------------- - ------------------------- + -------------------------------------------------- + --------------------- - --------------------------------- + ---------------------------------
2/3 2 9*(-1 + x) 8/3 2/3 2 2/3 2 5/3 3*(-1 + x)*(2 + x) 5/3 3 _________ 3*(-1 + x)*(2 + x) 5/3 2/3 3 _________ 2/3 3 _________
(-1 + x) *(2 + x) 3*(-1 + x) 3*(-1 + x) (-1 + x) *(2 + x) (-1 + x) *(2 + x) 3*(-1 + x) *(2 + x) 3*(-1 + x) *\/ |2 + x| 3*(-1 + x) *(2 + x) 3*(-1 + x) *(2 + x)*\/ |2 + x| 3*(-1 + x) *(2 + x)*\/ |2 + x|
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 + x
$$\frac{\frac{2 x \left(\frac{6 \sqrt[3]{x - 1} \delta\left(x + 2\right)}{\sqrt[3]{\left|{x + 2}\right|}} - \frac{\sqrt[3]{x - 1} \operatorname{sign}^{2}{\left(x + 2 \right)}}{\left|{x + 2}\right|^{\frac{4}{3}}} + \frac{2 \operatorname{sign}{\left(x + 2 \right)}}{\left(x - 1\right)^{\frac{2}{3}} \sqrt[3]{\left|{x + 2}\right|}} - \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{5}{3}}}\right)}{9 \left(x - 1\right)} - \frac{2 x \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right) \left(x + 2\right)} - \frac{x \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right)^{2}} - \frac{4 x \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right) \sqrt[3]{\left|{x + 2}\right|}} + \frac{6 x \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)^{2}} - \frac{2 x \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{5}{3}} \sqrt[3]{\left|{x + 2}\right|}} + \frac{10 x \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{5}{3}} \left(x + 2\right)} + \frac{5 x \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{8}{3}}} + \frac{2 \left(x + 1\right) \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right) \left(x + 2\right)} + \frac{4 \left(x + 1\right) \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right) \sqrt[3]{\left|{x + 2}\right|}} - \frac{8 \left(x + 1\right) \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)^{2}} + \frac{2 \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)} - \frac{10 \left(x + 1\right) \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{5}{3}} \left(x + 2\right)}}{x + 2}$$