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How to use it?
- Derivative of:
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Derivative of 1/(1+x^2)
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Derivative of e^x/x
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Derivative of -1/x
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Derivative of x^(1/3)
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Identical expressions
- arcctg(x+ two)*ln(cos(x^ two + one))
- arcctg(x plus 2) multiply by ln( co sinus of e of (x squared plus 1))
- arcctg(x plus two) multiply by ln( co sinus of e of (x to the power of two plus one))
- arcctg(x+2)*ln(cos(x2+1))
- arcctgx+2*lncosx2+1
- arcctg(x+2)*ln(cos(x²+1))
- arcctg(x+2)*ln(cos(x to the power of 2+1))
- arcctg(x+2)ln(cos(x^2+1))
- arcctg(x+2)ln(cos(x2+1))
- arcctgx+2lncosx2+1
- arcctgx+2lncosx^2+1
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Similar expressions
- arcctg(x+2)*ln(cos(x^2-1))
- arcctg(x-2)*ln(cos(x^2+1))
Derivative of arcctg(x+2)*ln(cos(x^2+1))
The solution
You have entered
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/ / 2 \\ acot(x + 2)*log\cos\x + 1//
$$\log{\left(\cos{\left(x^{2} + 1 \right)} \right)} \operatorname{acot}{\left(x + 2 \right)}$$
d / / / 2 \\\ --\acot(x + 2)*log\cos\x + 1/// dx
$$\frac{d}{d x} \log{\left(\cos{\left(x^{2} + 1 \right)} \right)} \operatorname{acot}{\left(x + 2 \right)}$$
The first derivative
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/ / 2 \\ / 2 \
log\cos\x + 1// 2*x*acot(x + 2)*sin\x + 1/
- ---------------- - ---------------------------
2 / 2 \
1 + (x + 2) cos\x + 1/
$$- \frac{2 x \sin{\left(x^{2} + 1 \right)} \operatorname{acot}{\left(x + 2 \right)}}{\cos{\left(x^{2} + 1 \right)}} - \frac{\log{\left(\cos{\left(x^{2} + 1 \right)} \right)}}{\left(x + 2\right)^{2} + 1}$$
The second derivative
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/ / / 2\ 2 2/ 2\\ / / 2\\ / 2\ \ | | 2 sin\1 + x / 2*x *sin \1 + x /| (2 + x)*log\cos\1 + x // 2*x*sin\1 + x / | 2*|- |2*x + ----------- + -----------------|*acot(2 + x) + ------------------------ + --------------------------| | | / 2\ 2/ 2\ | 2 / 2\ / 2\| | \ cos\1 + x / cos \1 + x / / / 2\ \1 + (2 + x) /*cos\1 + x /| \ \1 + (2 + x) / /
$$2 \cdot \left(\frac{2 x \sin{\left(x^{2} + 1 \right)}}{\left(\left(x + 2\right)^{2} + 1\right) \cos{\left(x^{2} + 1 \right)}} + \frac{\left(x + 2\right) \log{\left(\cos{\left(x^{2} + 1 \right)} \right)}}{\left(\left(x + 2\right)^{2} + 1\right)^{2}} - \left(\frac{2 x^{2} \sin^{2}{\left(x^{2} + 1 \right)}}{\cos^{2}{\left(x^{2} + 1 \right)}} + 2 x^{2} + \frac{\sin{\left(x^{2} + 1 \right)}}{\cos{\left(x^{2} + 1 \right)}}\right) \operatorname{acot}{\left(x + 2 \right)}\right)$$
The third derivative
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/ / / 2\ 2 2/ 2\\ / 2 \ \ | | 2 sin\1 + x / 2*x *sin \1 + x /| | 4*(2 + x) | / / 2\\ | |3*|2*x + ----------- + -----------------| |-1 + ------------|*log\cos\1 + x // | | | / 2\ 2/ 2\ | | 2| / 2/ 2\ 2 / 2\ 2 3/ 2\\ / 2\ | | \ cos\1 + x / cos \1 + x / / \ 1 + (2 + x) / | 3*sin \1 + x / 4*x *sin\1 + x / 4*x *sin \1 + x /| 6*x*(2 + x)*sin\1 + x / | 2*|------------------------------------------ - ------------------------------------ - 2*x*|3 + -------------- + ---------------- + -----------------|*acot(2 + x) - ---------------------------| | 2 2 | 2/ 2\ / 2\ 3/ 2\ | 2 | | 1 + (2 + x) / 2\ \ cos \1 + x / cos\1 + x / cos \1 + x / / / 2\ / 2\| \ \1 + (2 + x) / \1 + (2 + x) / *cos\1 + x //
$$2 \left(- \frac{6 x \left(x + 2\right) \sin{\left(x^{2} + 1 \right)}}{\left(\left(x + 2\right)^{2} + 1\right)^{2} \cos{\left(x^{2} + 1 \right)}} - 2 x \left(\frac{4 x^{2} \sin^{3}{\left(x^{2} + 1 \right)}}{\cos^{3}{\left(x^{2} + 1 \right)}} + \frac{4 x^{2} \sin{\left(x^{2} + 1 \right)}}{\cos{\left(x^{2} + 1 \right)}} + \frac{3 \sin^{2}{\left(x^{2} + 1 \right)}}{\cos^{2}{\left(x^{2} + 1 \right)}} + 3\right) \operatorname{acot}{\left(x + 2 \right)} - \frac{\left(\frac{4 \left(x + 2\right)^{2}}{\left(x + 2\right)^{2} + 1} - 1\right) \log{\left(\cos{\left(x^{2} + 1 \right)} \right)}}{\left(\left(x + 2\right)^{2} + 1\right)^{2}} + \frac{3 \cdot \left(\frac{2 x^{2} \sin^{2}{\left(x^{2} + 1 \right)}}{\cos^{2}{\left(x^{2} + 1 \right)}} + 2 x^{2} + \frac{\sin{\left(x^{2} + 1 \right)}}{\cos{\left(x^{2} + 1 \right)}}\right)}{\left(x + 2\right)^{2} + 1}\right)$$
The graph