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- Derivative of:
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Derivative of log
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Derivative of 5x
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Derivative of 1/(x-1)
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Derivative of 4*sqrt(x)
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Identical expressions
- arccos^ four (x)ln(x^ two +x− one)
- arc co sinus of e of to the power of 4(x)ln(x squared plus x−1)
- arc co sinus of e of to the power of four (x)ln(x to the power of two plus x− one)
- arccos4(x)ln(x2+x−1)
- arccos4xlnx2+x−1
- arccos⁴(x)ln(x²+x−1)
- arccos to the power of 4(x)ln(x to the power of 2+x−1)
- arccos^4xlnx^2+x−1
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Similar expressions
- arccos^4(x)ln(x^2-x−1)
Derivative of arccos^4(x)ln(x^2+x−1)
The solution
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[src]
4 / 2 \ acos (x)*log\x + x - 1/
$$\log{\left(\left(x^{2} + x\right) - 1 \right)} \operatorname{acos}^{4}{\left(x \right)}$$
acos(x)^4*log(x^2 + x - 1)
The first derivative
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4 3 / 2 \
acos (x)*(1 + 2*x) 4*acos (x)*log\x + x - 1/
------------------ - --------------------------
2 ________
x + x - 1 / 2
\/ 1 - x
$$\frac{\left(2 x + 1\right) \operatorname{acos}^{4}{\left(x \right)}}{\left(x^{2} + x\right) - 1} - \frac{4 \log{\left(\left(x^{2} + x\right) - 1 \right)} \operatorname{acos}^{3}{\left(x \right)}}{\sqrt{1 - x^{2}}}$$
The second derivative
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/ / 2\ \
| 2 | (1 + 2*x) | |
| acos (x)*|-2 + -----------| |
| | 2| |
2 | / 3 x*acos(x) \ / 2\ \ -1 + x + x / 8*(1 + 2*x)*acos(x) |
-acos (x)*|4*|------- + -----------|*log\-1 + x + x / + --------------------------- + -------------------------|
| | 2 3/2| 2 ________ |
| |-1 + x / 2\ | -1 + x + x / 2 / 2\|
\ \ \1 - x / / \/ 1 - x *\-1 + x + x //
$$- \left(\frac{\left(\frac{\left(2 x + 1\right)^{2}}{x^{2} + x - 1} - 2\right) \operatorname{acos}^{2}{\left(x \right)}}{x^{2} + x - 1} + 4 \left(\frac{x \operatorname{acos}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{3}{x^{2} - 1}\right) \log{\left(x^{2} + x - 1 \right)} + \frac{8 \left(2 x + 1\right) \operatorname{acos}{\left(x \right)}}{\sqrt{1 - x^{2}} \left(x^{2} + x - 1\right)}\right) \operatorname{acos}^{2}{\left(x \right)}$$
The third derivative
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/ / 2\ / 3 x*acos(x) \ / 2\\ | 3 | (1 + 2*x) | 6*(1 + 2*x)*|------- + -----------|*acos(x) 2 | (1 + 2*x) || | acos (x)*(1 + 2*x)*|-3 + -----------| | 2 3/2| 6*acos (x)*|-2 + -----------|| | / 2 2 2 \ | 2| |-1 + x / 2\ | | 2|| | | 6 acos (x) 9*x*acos(x) 3*x *acos (x)| / 2\ \ -1 + x + x / \ \1 - x / / \ -1 + x + x /| 2*|- 2*|----------- + ----------- - ----------- + -------------|*log\-1 + x + x / + ------------------------------------- - ------------------------------------------- + -----------------------------|*acos(x) | | 3/2 3/2 2 5/2 | 2 2 ________ | | |/ 2\ / 2\ / 2\ / 2\ | / 2\ -1 + x + x / 2 / 2\ | \ \\1 - x / \1 - x / \-1 + x / \1 - x / / \-1 + x + x / \/ 1 - x *\-1 + x + x / /
$$2 \left(\frac{\left(2 x + 1\right) \left(\frac{\left(2 x + 1\right)^{2}}{x^{2} + x - 1} - 3\right) \operatorname{acos}^{3}{\left(x \right)}}{\left(x^{2} + x - 1\right)^{2}} - \frac{6 \left(2 x + 1\right) \left(\frac{x \operatorname{acos}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{3}{x^{2} - 1}\right) \operatorname{acos}{\left(x \right)}}{x^{2} + x - 1} - 2 \left(\frac{3 x^{2} \operatorname{acos}^{2}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} - \frac{9 x \operatorname{acos}{\left(x \right)}}{\left(x^{2} - 1\right)^{2}} + \frac{\operatorname{acos}^{2}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{6}{\left(1 - x^{2}\right)^{\frac{3}{2}}}\right) \log{\left(x^{2} + x - 1 \right)} + \frac{6 \left(\frac{\left(2 x + 1\right)^{2}}{x^{2} + x - 1} - 2\right) \operatorname{acos}^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \left(x^{2} + x - 1\right)}\right) \operatorname{acos}{\left(x \right)}$$