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- Derivative of a implicit defined function
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How to use it?
- Derivative of:
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Derivative of cos(x/3)
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Derivative of √x
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Derivative of xcos(x)
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Derivative of -x^3+(1/2)*x^2-x+1
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Identical expressions
- cbrt((x^ three + one)/(3x- two))
- cubic root of ((x cubed plus 1) divide by (3x minus 2))
- cubic root of ((x to the power of three plus one) divide by (3x minus two))
- cbrt((x3+1)/(3x-2))
- cbrtx3+1/3x-2
- cbrt((x³+1)/(3x-2))
- cbrt((x to the power of 3+1)/(3x-2))
- cbrtx^3+1/3x-2
- cbrt((x^3+1) divide by (3x-2))
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Similar expressions
- cbrt((x^3-1)/(3x-2))
- cbrt((x^3+1)/(3x+2))
Derivative of cbrt((x^3+1)/(3x-2))
The solution
You have entered
[src]
_________
/ 3
/ x + 1
3 / -------
\/ 3*x - 2
$$\sqrt[3]{\frac{x^{3} + 1}{3 x - 2}}$$
((x^3 + 1)/(3*x - 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 quotient rule, which is:
and .
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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The derivative of a constant times a function is the constant times the derivative of the function.
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Apply the power rule: goes to
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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The result of the chain rule is:
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Now simplify:
The answer is:
The first derivative
[src]
_________
/ 3 / 2 3 \
/ x + 1 | x x + 1 |
3 / ------- *(3*x - 2)*|------- - ----------|
\/ 3*x - 2 |3*x - 2 2|
\ (3*x - 2) /
-----------------------------------------------
3
x + 1
$$\frac{\sqrt[3]{\frac{x^{3} + 1}{3 x - 2}} \left(3 x - 2\right) \left(\frac{x^{2}}{3 x - 2} - \frac{x^{3} + 1}{\left(3 x - 2\right)^{2}}\right)}{x^{3} + 1}$$
The second derivative
[src]
/ 2 \
| / 3 \ / 3 \ / 3 \|
__________ | | 2 1 + x | | 2 1 + x | 2 | 2 1 + x ||
/ 3 | |x - --------| 2 3*|x - --------| / 3\ 3*x *|x - --------||
/ 1 + x | \ -2 + 3*x/ 6*x \ -2 + 3*x/ 6*\1 + x / \ -2 + 3*x/|
3 / -------- *|2*x + ---------------- - -------- + ----------------- + ----------- - --------------------|
\/ -2 + 3*x | 3 -2 + 3*x -2 + 3*x 2 3 |
\ 1 + x (-2 + 3*x) 1 + x /
------------------------------------------------------------------------------------------------------------
3
1 + x
$$\frac{\sqrt[3]{\frac{x^{3} + 1}{3 x - 2}} \left(- \frac{3 x^{2} \left(x^{2} - \frac{x^{3} + 1}{3 x - 2}\right)}{x^{3} + 1} - \frac{6 x^{2}}{3 x - 2} + 2 x + \frac{\left(x^{2} - \frac{x^{3} + 1}{3 x - 2}\right)^{2}}{x^{3} + 1} + \frac{3 \left(x^{2} - \frac{x^{3} + 1}{3 x - 2}\right)}{3 x - 2} + \frac{6 \left(x^{3} + 1\right)}{\left(3 x - 2\right)^{2}}\right)}{x^{3} + 1}$$
The third derivative
[src]
/ 3 / 2 / 3\\ / 2 / 3\\ 2 / 3 \ / 2 / 3\\ 2 \
| / 3 \ | 3*x 3*\1 + x /| 2 | 3*x 3*\1 + x /| / 3 \ / 3 \ | 2 1 + x | | 3*x 3*\1 + x /| / 3 \ / 3 \ / 3 \|
__________ | | 2 1 + x | 12*|x - -------- + -----------| 12*x *|x - -------- + -----------| 2 | 2 1 + x | | 2 1 + x | 6*|x - --------|*|x - -------- + -----------| | 2 1 + x | 4 | 2 1 + x | 2 | 2 1 + x ||
/ 3 | |x - --------| / 3\ | -2 + 3*x 2| 2 | -2 + 3*x 2| 9*x *|x - --------| 6*x*|x - --------| \ -2 + 3*x/ | -2 + 3*x 2| 9*|x - --------| 18*x *|x - --------| 18*x *|x - --------||
/ 1 + x | \ -2 + 3*x/ 54*\1 + x / 18*x \ (-2 + 3*x) / 54*x \ (-2 + 3*x) / \ -2 + 3*x/ \ -2 + 3*x/ \ (-2 + 3*x) / \ -2 + 3*x/ \ -2 + 3*x/ \ -2 + 3*x/|
3 / -------- *|2 + ---------------- - ----------- - -------- + ------------------------------- + ----------- - ---------------------------------- - --------------------- - ------------------- + ---------------------------------------------- + ------------------- + --------------------- - ---------------------|
\/ -2 + 3*x | 2 3 -2 + 3*x -2 + 3*x 2 3 2 3 3 / 3\ 2 / 3\ |
| / 3\ (-2 + 3*x) (-2 + 3*x) 1 + x / 3\ 1 + x 1 + x \1 + x /*(-2 + 3*x) / 3\ \1 + x /*(-2 + 3*x) |
\ \1 + x / \1 + x / \1 + x / /
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3
1 + x
$$\frac{\sqrt[3]{\frac{x^{3} + 1}{3 x - 2}} \left(\frac{18 x^{4} \left(x^{2} - \frac{x^{3} + 1}{3 x - 2}\right)}{\left(x^{3} + 1\right)^{2}} - \frac{9 x^{2} \left(x^{2} - \frac{x^{3} + 1}{3 x - 2}\right)^{2}}{\left(x^{3} + 1\right)^{2}} - \frac{12 x^{2} \left(- \frac{3 x^{2}}{3 x - 2} + x + \frac{3 \left(x^{3} + 1\right)}{\left(3 x - 2\right)^{2}}\right)}{x^{3} + 1} - \frac{18 x^{2} \left(x^{2} - \frac{x^{3} + 1}{3 x - 2}\right)}{\left(3 x - 2\right) \left(x^{3} + 1\right)} + \frac{54 x^{2}}{\left(3 x - 2\right)^{2}} - \frac{6 x \left(x^{2} - \frac{x^{3} + 1}{3 x - 2}\right)}{x^{3} + 1} - \frac{18 x}{3 x - 2} + \frac{\left(x^{2} - \frac{x^{3} + 1}{3 x - 2}\right)^{3}}{\left(x^{3} + 1\right)^{2}} + \frac{6 \left(x^{2} - \frac{x^{3} + 1}{3 x - 2}\right) \left(- \frac{3 x^{2}}{3 x - 2} + x + \frac{3 \left(x^{3} + 1\right)}{\left(3 x - 2\right)^{2}}\right)}{x^{3} + 1} + 2 + \frac{9 \left(x^{2} - \frac{x^{3} + 1}{3 x - 2}\right)^{2}}{\left(3 x - 2\right) \left(x^{3} + 1\right)} + \frac{12 \left(- \frac{3 x^{2}}{3 x - 2} + x + \frac{3 \left(x^{3} + 1\right)}{\left(3 x - 2\right)^{2}}\right)}{3 x - 2} - \frac{54 \left(x^{3} + 1\right)}{\left(3 x - 2\right)^{3}}\right)}{x^{3} + 1}$$