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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 (x+3)^2*(x+5)-1
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Derivative of 1/(x-3)
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Derivative of x^6*e^x
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Derivative of -x^5+2*x^3-3*x^2-1
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Identical expressions
- (arcsinx/x)^(one /x^ two)
- (arc sinus of x divide by x) to the power of (1 divide by x squared )
- (arc sinus of x divide by x) to the power of (one divide by x to the power of two)
- (arcsinx/x)(1/x2)
- arcsinx/x1/x2
- (arcsinx/x)^(1/x²)
- (arcsinx/x) to the power of (1/x to the power of 2)
- arcsinx/x^1/x^2
- (arcsinx divide by x)^(1 divide by x^2)
Derivative of (arcsinx/x)^(1/x^2)
The solution
You have entered
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1
1*--
2
x
/asin(x)\
|-------|
\ x /
$$\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)^{1 \cdot \frac{1}{x^{2}}}$$
/ 1 \ | 1*--| | 2| | x | d |/asin(x)\ | --||-------| | dx\\ x / /
$$\frac{d}{d x} \left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)^{1 \cdot \frac{1}{x^{2}}}$$
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
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Now simplify:
The answer is:
The first derivative
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1 / 1 asin(x)\
-- | ------------- - -------|
2 | /asin(x)\ ________ 2 |
x | 2*log|-------| / 2 x |
/asin(x)\ | \ x / x*\/ 1 - x |
|-------| *|- -------------- + -----------------------|
\ x / | 3 x*asin(x) |
\ x /
$$\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)^{\frac{1}{x^{2}}} \left(\frac{\frac{1}{x \sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x^{2}}}{x \operatorname{asin}{\left(x \right)}} - \frac{2 \log{\left(\frac{\operatorname{asin}{\left(x \right)}}{x} \right)}}{x^{3}}\right)$$
The second derivative
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/ 2 \
|/ 1 asin(x) \ |
||----------- - ------- | |
1 || ________ x /asin(x)\| 1 2 2*asin(x) / 1 asin(x)\ 1 asin(x) |
-- || / 2 2*log|-------|| ----------- - -------------- + --------- 3*|----------- - -------| ----------- - ------- |
2 ||\/ 1 - x \ x /| 3/2 ________ 3 /asin(x)\ | ________ x | ________ x |
x ||--------------------- - --------------| / 2\ 2 / 2 x 6*log|-------| | / 2 | / 2 |
/asin(x)\ |\ asin(x) x / \1 - x / x *\/ 1 - x \ x / \\/ 1 - x / \/ 1 - x |
|-------| *|----------------------------------------- + ---------------------------------------- + -------------- - ------------------------- - ----------------------|
\ x / | 3 asin(x) 3 2 ________ |
| x x x *asin(x) / 2 2 |
\ x*\/ 1 - x *asin (x)/
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x
$$\frac{\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)^{\frac{1}{x^{2}}} \left(\frac{\frac{1}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{2}{x^{2} \sqrt{1 - x^{2}}} + \frac{2 \operatorname{asin}{\left(x \right)}}{x^{3}}}{\operatorname{asin}{\left(x \right)}} - \frac{\frac{1}{\sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x}}{x \sqrt{1 - x^{2}} \operatorname{asin}^{2}{\left(x \right)}} - \frac{3 \cdot \left(\frac{1}{\sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x}\right)}{x^{2} \operatorname{asin}{\left(x \right)}} + \frac{\left(\frac{\frac{1}{\sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x}}{\operatorname{asin}{\left(x \right)}} - \frac{2 \log{\left(\frac{\operatorname{asin}{\left(x \right)}}{x} \right)}}{x}\right)^{2}}{x^{3}} + \frac{6 \log{\left(\frac{\operatorname{asin}{\left(x \right)}}{x} \right)}}{x^{3}}\right)}{x}$$
The third derivative
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/ / 1 asin(x) \ / 1 2 2*asin(x) / 1 asin(x)\ 1 asin(x) \ \
| 3 |----------- - ------- | |----------- - -------------- + --------- 3*|----------- - -------| ----------- - ------- | |
|/ 1 asin(x) \ | ________ x /asin(x)\| | 3/2 ________ 3 /asin(x)\ | ________ x | ________ x | |
||----------- - ------- | | / 2 2*log|-------|| |/ 2\ 2 / 2 x 6*log|-------| | / 2 | / 2 | |
1 || ________ x /asin(x)\| 6*asin(x) 2 3*x 6 1 asin(x) / 1 2 2*asin(x)\ / 1 2 2*asin(x)\ |\/ 1 - x \ x /| |\1 - x / x *\/ 1 - x \ x / \\/ 1 - x / \/ 1 - x | / 1 asin(x)\ / 1 asin(x)\ / 1 asin(x)\|
-- || / 2 2*log|-------|| - --------- - ------------- + ----------- + -------------- ----------- - ------- 4*|----------- - -------------- + ---------| 2*|----------- - -------------- + ---------| 3*|--------------------- - --------------|*|---------------------------------------- + -------------- - ------------------------- - ----------------------| 12*|----------- - -------| 2*|----------- - -------| 4*|----------- - -------||
2 ||\/ 1 - x \ x /| 4 3/2 5/2 ________ /asin(x)\ ________ x | 3/2 ________ 3 | | 3/2 ________ 3 | \ asin(x) x / | asin(x) 3 2 ________ | | ________ x | | ________ x | | ________ x ||
x ||--------------------- - --------------| x / 2\ / 2\ 3 / 2 24*log|-------| / 2 |/ 2\ 2 / 2 x | |/ 2\ 2 / 2 x | | x x *asin(x) / 2 2 | | / 2 | | / 2 | | / 2 ||
/asin(x)\ |\ asin(x) x / x*\1 - x / \1 - x / x *\/ 1 - x \ x / \/ 1 - x \\1 - x / x *\/ 1 - x / \\1 - x / x *\/ 1 - x / \ x*\/ 1 - x *asin (x)/ \\/ 1 - x / \\/ 1 - x / \\/ 1 - x /|
|-------| *|----------------------------------------- + ---------------------------------------------------------- - --------------- - --------------------- - -------------------------------------------- - -------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------- - ------------------------- + -------------------------|
\ x / | 5 asin(x) 4 3/2 x*asin(x) ________ 2 3 / 2\ 3 ________ |
| x x / 2\ 2 / 2 2 x x *asin(x) x*\-1 + x /*asin (x) 2 / 2 2 |
\ \1 - x / *asin (x) \/ 1 - x *asin (x) x *\/ 1 - x *asin (x) /
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x
$$\frac{\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)^{\frac{1}{x^{2}}} \left(\frac{\frac{3 x}{\left(1 - x^{2}\right)^{\frac{5}{2}}} - \frac{2}{x \left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{6}{x^{3} \sqrt{1 - x^{2}}} - \frac{6 \operatorname{asin}{\left(x \right)}}{x^{4}}}{\operatorname{asin}{\left(x \right)}} - \frac{2 \left(\frac{1}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{2}{x^{2} \sqrt{1 - x^{2}}} + \frac{2 \operatorname{asin}{\left(x \right)}}{x^{3}}\right)}{\sqrt{1 - x^{2}} \operatorname{asin}^{2}{\left(x \right)}} - \frac{\frac{1}{\sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x}}{\left(1 - x^{2}\right)^{\frac{3}{2}} \operatorname{asin}^{2}{\left(x \right)}} - \frac{4 \left(\frac{1}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{2}{x^{2} \sqrt{1 - x^{2}}} + \frac{2 \operatorname{asin}{\left(x \right)}}{x^{3}}\right)}{x \operatorname{asin}{\left(x \right)}} - \frac{2 \cdot \left(\frac{1}{\sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x}\right)}{x \left(x^{2} - 1\right) \operatorname{asin}^{3}{\left(x \right)}} + \frac{3 \left(\frac{\frac{1}{\sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x}}{\operatorname{asin}{\left(x \right)}} - \frac{2 \log{\left(\frac{\operatorname{asin}{\left(x \right)}}{x} \right)}}{x}\right) \left(\frac{\frac{1}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{2}{x^{2} \sqrt{1 - x^{2}}} + \frac{2 \operatorname{asin}{\left(x \right)}}{x^{3}}}{\operatorname{asin}{\left(x \right)}} - \frac{\frac{1}{\sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x}}{x \sqrt{1 - x^{2}} \operatorname{asin}^{2}{\left(x \right)}} - \frac{3 \cdot \left(\frac{1}{\sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x}\right)}{x^{2} \operatorname{asin}{\left(x \right)}} + \frac{6 \log{\left(\frac{\operatorname{asin}{\left(x \right)}}{x} \right)}}{x^{3}}\right)}{x^{2}} + \frac{4 \cdot \left(\frac{1}{\sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x}\right)}{x^{2} \sqrt{1 - x^{2}} \operatorname{asin}^{2}{\left(x \right)}} + \frac{12 \cdot \left(\frac{1}{\sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x}\right)}{x^{3} \operatorname{asin}{\left(x \right)}} - \frac{24 \log{\left(\frac{\operatorname{asin}{\left(x \right)}}{x} \right)}}{x^{4}} + \frac{\left(\frac{\frac{1}{\sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x}}{\operatorname{asin}{\left(x \right)}} - \frac{2 \log{\left(\frac{\operatorname{asin}{\left(x \right)}}{x} \right)}}{x}\right)^{3}}{x^{5}}\right)}{x}$$
The graph