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How to use it?
- Derivative of:
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Derivative of 2*x/(x^2+4)
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Derivative of sqrt(x)-1/(sqrt(x)*2-x-1)
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Derivative of ln(x^2-4x)
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Derivative of cos(pi/3-4*x)
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Identical expressions
- arcsin((sqrt(x)))^((x+ four)/x)
- arc sinus of (( square root of (x))) to the power of ((x plus 4) divide by x)
- arc sinus of (( square root of (x))) to the power of ((x plus four) divide by x)
- arcsin((√(x)))^((x+4)/x)
- arcsin((sqrt(x)))((x+4)/x)
- arcsinsqrtxx+4/x
- arcsinsqrtx^x+4/x
- arcsin((sqrt(x)))^((x+4) divide by x)
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Similar expressions
- arcsin((sqrt(x)))^((x-4)/x)
Derivative of arcsin((sqrt(x)))^((x+4)/x)
The solution
You have entered
[src]
x + 4
-----
x
/ / ___\\
\asin\\/ x //
$$\operatorname{asin}^{\frac{x + 4}{x}}{\left(\sqrt{x} \right)}$$
asin(sqrt(x))^((x + 4)/x)
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
[src]
x + 4
-----
x
/ / ___\\ //1 x + 4\ / / ___\\ x + 4 \
\asin\\/ x // *||- - -----|*log\asin\\/ x // + ----------------------------|
||x 2 | 3/2 _______ / ___\|
\\ x / 2*x *\/ 1 - x *asin\\/ x //
$$\left(\left(\frac{1}{x} - \frac{x + 4}{x^{2}}\right) \log{\left(\operatorname{asin}{\left(\sqrt{x} \right)} \right)} + \frac{x + 4}{2 x^{\frac{3}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}}\right) \operatorname{asin}^{\frac{x + 4}{x}}{\left(\sqrt{x} \right)}$$
The second derivative
[src]
/ 2 \
|/ / 4 + x\ / / ___\\ \ |
||2*|1 - -----|*log\asin\\/ x // | |
4 + x || \ x / 4 + x | |
----- ||------------------------------ + --------------------------| / 4 + x\ / / ___\\ 4 + x |
x || x 3/2 _______ / ___\| 2*|1 - -----|*log\asin\\/ x // 1 - ----- |
/ / ___\\ |\ x *\/ 1 - x *asin\\/ x // 1 \ x / x 3*(4 + x) 4 + x 4 + x |
\asin\\/ x // *|-------------------------------------------------------------- + ---------------------------- - ------------------------------ + ---------------------------- - ---------------------------- + -------------------------- + -----------------------------|
| 4 3/2 _______ / ___\ 2 3/2 _______ / ___\ 5/2 _______ / ___\ 2 2/ ___\ 3/2 3/2 / ___\|
\ 2*x *\/ 1 - x *asin\\/ x / x 2*x *\/ 1 - x *asin\\/ x / 4*x *\/ 1 - x *asin\\/ x / 4*x *(-1 + x)*asin \\/ x / 4*x *(1 - x) *asin\\/ x //
$$\left(\frac{\left(\frac{2 \left(1 - \frac{x + 4}{x}\right) \log{\left(\operatorname{asin}{\left(\sqrt{x} \right)} \right)}}{x} + \frac{x + 4}{x^{\frac{3}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}}\right)^{2}}{4} - \frac{2 \left(1 - \frac{x + 4}{x}\right) \log{\left(\operatorname{asin}{\left(\sqrt{x} \right)} \right)}}{x^{2}} + \frac{x + 4}{4 x^{2} \left(x - 1\right) \operatorname{asin}^{2}{\left(\sqrt{x} \right)}} + \frac{1 - \frac{x + 4}{x}}{2 x^{\frac{3}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}} + \frac{1}{2 x^{\frac{3}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}} + \frac{x + 4}{4 x^{\frac{3}{2}} \left(1 - x\right)^{\frac{3}{2}} \operatorname{asin}{\left(\sqrt{x} \right)}} - \frac{3 \left(x + 4\right)}{4 x^{\frac{5}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}}\right) \operatorname{asin}^{\frac{x + 4}{x}}{\left(\sqrt{x} \right)}$$
The third derivative
[src]
/ 3 \
|/ / 4 + x\ / / ___\\ \ / / 4 + x\ / / ___\\ \ / / 4 + x\ / / ___\\ / 4 + x\ \ |
||2*|1 - -----|*log\asin\\/ x // | |2*|1 - -----|*log\asin\\/ x // | | 8*|1 - -----|*log\asin\\/ x // 2*|1 - -----| | |
4 + x || \ x / 4 + x | | \ x / 4 + x | | \ x / 2 4 + x 4 + x 3*(4 + x) \ x / | |
----- ||------------------------------ + --------------------------| 3*|------------------------------ + --------------------------|*|- ------------------------------ + -------------------------- + ------------------------ + --------------------------- - -------------------------- + --------------------------| / 4 + x\ / / ___\\ / 4 + x\ 4 + x 4 + x |
x || x 3/2 _______ / ___\| | x 3/2 _______ / ___\| | 2 3/2 _______ / ___\ 2 2/ ___\ 3/2 3/2 / ___\ 5/2 _______ / ___\ 3/2 _______ / ___\| 6*|1 - -----|*log\asin\\/ x // 9*|1 - -----| 1 - ----- 1 - ----- |
/ / ___\\ |\ x *\/ 1 - x *asin\\/ x // \ x *\/ 1 - x *asin\\/ x // \ x x *\/ 1 - x *asin\\/ x / x *(-1 + x)*asin \\/ x / x *(1 - x) *asin\\/ x / x *\/ 1 - x *asin\\/ x / x *\/ 1 - x *asin\\/ x // 1 1 \ x / 3 \ x / 7*(4 + x) 3*(4 + x) 3*(4 + x) x 4 + x x 3*(4 + x) 15*(4 + x) |
\asin\\/ x // *|-------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------- + ----------------------------- + ------------------------------ - ---------------------------- - ---------------------------- - -------------------------- - ----------------------------- - --------------------------- + -------------------------- + ------------------------------ + ----------------------------- + ----------------------------- + ----------------------------|
| 8 8 2 2/ ___\ 3/2 3/2 / ___\ 3 5/2 _______ / ___\ 5/2 _______ / ___\ 3 2/ ___\ 5/2 3/2 / ___\ 2 2 2/ ___\ 2 2/ ___\ 5/2 3/2 3/ ___\ 3/2 3/2 / ___\ 3/2 5/2 / ___\ 7/2 _______ / ___\|
\ 2*x *(-1 + x)*asin \\/ x / 2*x *(1 - x) *asin\\/ x / x 2*x *\/ 1 - x *asin\\/ x / 4*x *\/ 1 - x *asin\\/ x / 8*x *(-1 + x)*asin \\/ x / 4*x *(1 - x) *asin\\/ x / 8*x *(-1 + x) *asin \\/ x / 4*x *(-1 + x)*asin \\/ x / 4*x *(1 - x) *asin \\/ x / 4*x *(1 - x) *asin\\/ x / 8*x *(1 - x) *asin\\/ x / 8*x *\/ 1 - x *asin\\/ x //
$$\left(\frac{\left(\frac{2 \left(1 - \frac{x + 4}{x}\right) \log{\left(\operatorname{asin}{\left(\sqrt{x} \right)} \right)}}{x} + \frac{x + 4}{x^{\frac{3}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}}\right)^{3}}{8} + \frac{3 \left(\frac{2 \left(1 - \frac{x + 4}{x}\right) \log{\left(\operatorname{asin}{\left(\sqrt{x} \right)} \right)}}{x} + \frac{x + 4}{x^{\frac{3}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}}\right) \left(- \frac{8 \left(1 - \frac{x + 4}{x}\right) \log{\left(\operatorname{asin}{\left(\sqrt{x} \right)} \right)}}{x^{2}} + \frac{x + 4}{x^{2} \left(x - 1\right) \operatorname{asin}^{2}{\left(\sqrt{x} \right)}} + \frac{2 \left(1 - \frac{x + 4}{x}\right)}{x^{\frac{3}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}} + \frac{2}{x^{\frac{3}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}} + \frac{x + 4}{x^{\frac{3}{2}} \left(1 - x\right)^{\frac{3}{2}} \operatorname{asin}{\left(\sqrt{x} \right)}} - \frac{3 \left(x + 4\right)}{x^{\frac{5}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}}\right)}{8} + \frac{1 - \frac{x + 4}{x}}{4 x^{2} \left(x - 1\right) \operatorname{asin}^{2}{\left(\sqrt{x} \right)}} + \frac{1}{2 x^{2} \left(x - 1\right) \operatorname{asin}^{2}{\left(\sqrt{x} \right)}} - \frac{3 \left(x + 4\right)}{8 x^{2} \left(x - 1\right)^{2} \operatorname{asin}^{2}{\left(\sqrt{x} \right)}} + \frac{6 \left(1 - \frac{x + 4}{x}\right) \log{\left(\operatorname{asin}{\left(\sqrt{x} \right)} \right)}}{x^{3}} - \frac{7 \left(x + 4\right)}{8 x^{3} \left(x - 1\right) \operatorname{asin}^{2}{\left(\sqrt{x} \right)}} + \frac{1 - \frac{x + 4}{x}}{4 x^{\frac{3}{2}} \left(1 - x\right)^{\frac{3}{2}} \operatorname{asin}{\left(\sqrt{x} \right)}} + \frac{1}{2 x^{\frac{3}{2}} \left(1 - x\right)^{\frac{3}{2}} \operatorname{asin}{\left(\sqrt{x} \right)}} + \frac{3 \left(x + 4\right)}{8 x^{\frac{3}{2}} \left(1 - x\right)^{\frac{5}{2}} \operatorname{asin}{\left(\sqrt{x} \right)}} - \frac{9 \left(1 - \frac{x + 4}{x}\right)}{4 x^{\frac{5}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}} - \frac{3}{2 x^{\frac{5}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}} - \frac{3 \left(x + 4\right)}{4 x^{\frac{5}{2}} \left(1 - x\right)^{\frac{3}{2}} \operatorname{asin}{\left(\sqrt{x} \right)}} + \frac{x + 4}{4 x^{\frac{5}{2}} \left(1 - x\right)^{\frac{3}{2}} \operatorname{asin}^{3}{\left(\sqrt{x} \right)}} + \frac{15 \left(x + 4\right)}{8 x^{\frac{7}{2}} \sqrt{1 - x} \operatorname{asin}{\left(\sqrt{x} \right)}}\right) \operatorname{asin}^{\frac{x + 4}{x}}{\left(\sqrt{x} \right)}$$