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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 50/x
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Derivative of e^sin^2x
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Derivative of cos(x)-tan(x)
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Derivative of arsinx
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Identical expressions
- sqrtx^arsin(x)
- square root of x to the power of ar sinus of (x)
- √x^arsin(x)
- sqrtxarsin(x)
- sqrtxarsinx
- sqrtx^arsinx
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Similar expressions
- sqrtx^arsinx
Derivative of sqrtx^arsin(x)
The solution
You have entered
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asin(x) ___ \/ x
$$\left(\sqrt{x}\right)^{\operatorname{asin}{\left(x \right)}}$$
(sqrt(x))^asin(x)
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
The answer is:
The first derivative
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asin(x)
------- / / ___\ \
2 | log\\/ x / asin(x)|
x *|----------- + -------|
| ________ 2*x |
| / 2 |
\\/ 1 - x /
$$x^{\frac{\operatorname{asin}{\left(x \right)}}{2}} \left(\frac{\log{\left(\sqrt{x} \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{2 x}\right)$$
The second derivative
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/ / / ___\\ \
| /asin(x) log(x) \ |asin(x) 2*log\\/ x /| |
| |------- + -----------|*|------- + ------------| |
asin(x) | | x ________| | x ________ | |
------- | | / 2 | | / 2 | / ___\|
2 | 1 asin(x) \ \/ 1 - x / \ \/ 1 - x / x*log\\/ x /|
x *|------------- - ------- + ------------------------------------------------ + ------------|
| ________ 2 4 3/2 |
| / 2 2*x / 2\ |
\x*\/ 1 - x \1 - x / /
$$x^{\frac{\operatorname{asin}{\left(x \right)}}{2}} \left(\frac{x \log{\left(\sqrt{x} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{\left(\frac{2 \log{\left(\sqrt{x} \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{x}\right) \left(\frac{\log{\left(x \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{x}\right)}{4} + \frac{1}{x \sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{2 x^{2}}\right)$$
The third derivative
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/ / / ___\\ / / ___\\ 2 / / ___\\ \
| /asin(x) log(x) \ | asin(x) 2 2*x*log\\/ x /| |asin(x) 2*log\\/ x /| / asin(x) 2 x*log(x) \ /asin(x) log(x) \ |asin(x) 2*log\\/ x /| |
| |------- + -----------|*|- ------- + ------------- + --------------| |------- + ------------|*|- ------- + ------------- + -----------| |------- + -----------| *|------- + ------------| |
asin(x) | | x ________| | 2 ________ 3/2 | | x ________ | | 2 ________ 3/2| | x ________| | x ________ | |
------- | / ___\ | / 2 | | x / 2 / 2\ | | / 2 | | x / 2 / 2\ | | / 2 | | / 2 | 2 / ___\|
2 | 3 asin(x) log\\/ x / \ \/ 1 - x / \ x*\/ 1 - x \1 - x / / 3 \ \/ 1 - x / \ x*\/ 1 - x \1 - x / / \ \/ 1 - x / \ \/ 1 - x / 3*x *log\\/ x /|
x *|------------- + ------- + ----------- + -------------------------------------------------------------------- - ---------------- + ------------------------------------------------------------------ + ------------------------------------------------- + ---------------|
| 3/2 3 3/2 2 ________ 4 8 5/2 |
| / 2\ x / 2\ 2 / 2 / 2\ |
\2*\1 - x / \1 - x / 2*x *\/ 1 - x \1 - x / /
$$x^{\frac{\operatorname{asin}{\left(x \right)}}{2}} \left(\frac{3 x^{2} \log{\left(\sqrt{x} \right)}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} + \frac{\left(\frac{2 \log{\left(\sqrt{x} \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{x}\right) \left(\frac{\log{\left(x \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{x}\right)^{2}}{8} + \frac{\left(\frac{2 \log{\left(\sqrt{x} \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{x}\right) \left(\frac{x \log{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{2}{x \sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x^{2}}\right)}{4} + \frac{\left(\frac{\log{\left(x \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{x}\right) \left(\frac{2 x \log{\left(\sqrt{x} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{2}{x \sqrt{1 - x^{2}}} - \frac{\operatorname{asin}{\left(x \right)}}{x^{2}}\right)}{2} + \frac{\log{\left(\sqrt{x} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{3}{2 \left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{3}{2 x^{2} \sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{x^{3}}\right)$$