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- Derivative of:
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Derivative of 6
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Derivative of x^2+3x
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Derivative of sqrt(1-x)
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Derivative of ln(x)/x
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Identical expressions
- arcsin(ln(x))/ln(arcsin(x))
- arc sinus of (ln(x)) divide by ln(arc sinus of (x))
- arcsinlnx/lnarcsinx
- arcsin(ln(x)) divide by ln(arcsin(x))
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Similar expressions
- arcsin(ln(x))/ln(arcsinx)
Derivative of arcsin(ln(x))/ln(arcsin(x))
The solution
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[src]
asin(log(x)) ------------ log(asin(x))
$$\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{\log{\left(\operatorname{asin}{\left(x \right)} \right)}}$$
asin(log(x))/log(asin(x))
The first derivative
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1 asin(log(x))
------------------------------- - ---------------------------------
_____________ ________
/ 2 / 2 2
x*\/ 1 - log (x) *log(asin(x)) \/ 1 - x *asin(x)*log (asin(x))
$$- \frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{\sqrt{1 - x^{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)}^{2} \operatorname{asin}{\left(x \right)}} + \frac{1}{x \sqrt{1 - \log{\left(x \right)}^{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)}}$$
The second derivative
[src]
/ / x 1 2 \ \
| log(x) |----------- + ----------------- + ------------------------------|*asin(log(x)) |
| 1 + ------------ | 3/2 / 2\ / 2\ | |
| 2 |/ 2\ \-1 + x /*asin(x) \-1 + x /*asin(x)*log(asin(x))| |
| -1 + log (x) \\1 - x / / 2 |
-|------------------- + ------------------------------------------------------------------------------- + ---------------------------------------------------|
| _____________ asin(x)*log(asin(x)) ________ _____________ |
| 2 / 2 / 2 / 2 |
\x *\/ 1 - log (x) x*\/ 1 - x *\/ 1 - log (x) *asin(x)*log(asin(x))/
---------------------------------------------------------------------------------------------------------------------------------------------------------------
log(asin(x))
$$- \frac{\frac{\left(\frac{x}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{1}{\left(x^{2} - 1\right) \operatorname{asin}{\left(x \right)}} + \frac{2}{\left(x^{2} - 1\right) \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}}\right) \operatorname{asin}{\left(\log{\left(x \right)} \right)}}{\log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} + \frac{2}{x \sqrt{1 - x^{2}} \sqrt{1 - \log{\left(x \right)}^{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} + \frac{1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}}{x^{2} \sqrt{1 - \log{\left(x \right)}^{2}}}}{\log{\left(\operatorname{asin}{\left(x \right)} \right)}}$$
The third derivative
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2 / 2 \
1 3*log(x) 3*log (x) | 1 2 3*x 3*x 6 6 6*x | / x 1 2 \
2 - ------------ + ------------ + --------------- |----------- + -------------------- + ----------- - ------------------ + --------------------------------- + ---------------------------------- - -------------------------------|*asin(log(x)) 3*|----------- + ----------------- + ------------------------------| / log(x) \
2 2 2 | 3/2 3/2 5/2 2 3/2 3/2 2 | | 3/2 / 2\ / 2\ | 3*|1 + ------------|
-1 + log (x) -1 + log (x) / 2 \ |/ 2\ / 2\ 2 / 2\ / 2\ / 2\ 2 / 2\ 2 2 / 2\ | |/ 2\ \-1 + x /*asin(x) \-1 + x /*asin(x)*log(asin(x))| | 2 |
\-1 + log (x)/ \\1 - x / \1 - x / *asin (x) \1 - x / \-1 + x / *asin(x) \1 - x / *asin (x)*log(asin(x)) \1 - x / *asin (x)*log (asin(x)) \-1 + x / *asin(x)*log(asin(x))/ \\1 - x / / \ -1 + log (x)/
------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------- + ----------------------------------------------------
_____________ asin(x)*log(asin(x)) _____________ ________ _____________
3 / 2 / 2 2 / 2 / 2
x *\/ 1 - log (x) x*\/ 1 - log (x) *asin(x)*log(asin(x)) x *\/ 1 - x *\/ 1 - log (x) *asin(x)*log(asin(x))
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
log(asin(x))
$$\frac{- \frac{\left(\frac{3 x^{2}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} - \frac{3 x}{\left(x^{2} - 1\right)^{2} \operatorname{asin}{\left(x \right)}} - \frac{6 x}{\left(x^{2} - 1\right)^{2} \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} + \frac{1}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{2}{\left(1 - x^{2}\right)^{\frac{3}{2}} \operatorname{asin}^{2}{\left(x \right)}} + \frac{6}{\left(1 - x^{2}\right)^{\frac{3}{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}^{2}{\left(x \right)}} + \frac{6}{\left(1 - x^{2}\right)^{\frac{3}{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)}^{2} \operatorname{asin}^{2}{\left(x \right)}}\right) \operatorname{asin}{\left(\log{\left(x \right)} \right)}}{\log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} - \frac{3 \left(\frac{x}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{1}{\left(x^{2} - 1\right) \operatorname{asin}{\left(x \right)}} + \frac{2}{\left(x^{2} - 1\right) \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}}\right)}{x \sqrt{1 - \log{\left(x \right)}^{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} + \frac{3 \left(1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}\right)}{x^{2} \sqrt{1 - x^{2}} \sqrt{1 - \log{\left(x \right)}^{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} + \frac{2 + \frac{3 \log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1} - \frac{1}{\log{\left(x \right)}^{2} - 1} + \frac{3 \log{\left(x \right)}^{2}}{\left(\log{\left(x \right)}^{2} - 1\right)^{2}}}{x^{3} \sqrt{1 - \log{\left(x \right)}^{2}}}}{\log{\left(\operatorname{asin}{\left(x \right)} \right)}}$$