Mister Exam
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- Derivative of:
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Derivative of x^-6
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Derivative of x^2*sqrt(1-x^2)
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Derivative of x^3*cot(x)
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Derivative of sin(x)+3
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Identical expressions
- (tg(sqrt(x)))^(ln(x))
- (tg( square root of (x))) to the power of (ln(x))
- (tg(√(x)))^(ln(x))
- (tg(sqrt(x)))(ln(x))
- tgsqrtxlnx
- tgsqrtx^lnx
Derivative of (tg(sqrt(x)))^(ln(x))
The solution
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log(x)/ ___\ tan \\/ x /
$$\tan^{\log{\left(x \right)}}{\left(\sqrt{x} \right)}$$
tan(sqrt(x))^log(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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/ / / ___\\ / 2/ ___\\ \
log(x)/ ___\ |log\tan\\/ x // \1 + tan \\/ x //*log(x)|
tan \\/ x /*|--------------- + ------------------------|
| x ___ / ___\ |
\ 2*\/ x *tan\\/ x / /
$$\left(\frac{\log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{2 \sqrt{x} \tan{\left(\sqrt{x} \right)}}\right) \tan^{\log{\left(x \right)}}{\left(\sqrt{x} \right)}$$
The second derivative
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/ 2 \
|/ / / ___\\ / 2/ ___\\ \ |
||2*log\tan\\/ x // \1 + tan \\/ x //*log(x)| |
||----------------- + ------------------------| 2 |
|| x ___ / ___\ | / / ___\\ 2/ ___\ / 2/ ___\\ / 2/ ___\\ / 2/ ___\\ |
log(x)/ ___\ |\ \/ x *tan\\/ x / / log\tan\\/ x // 1 + tan \\/ x / \1 + tan \\/ x //*log(x) \1 + tan \\/ x // *log(x) \1 + tan \\/ x //*log(x)|
tan \\/ x /*|----------------------------------------------- - --------------- + --------------- + ------------------------ - ------------------------- - ------------------------|
| 4 2 3/2 / ___\ 2*x 2/ ___\ 3/2 / ___\ |
\ x x *tan\\/ x / 4*x*tan \\/ x / 4*x *tan\\/ x / /
$$\left(\frac{\left(\frac{2 \log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{\sqrt{x} \tan{\left(\sqrt{x} \right)}}\right)^{2}}{4} - \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{2} \log{\left(x \right)}}{4 x \tan^{2}{\left(\sqrt{x} \right)}} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{2 x} - \frac{\log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x^{2}} - \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{4 x^{\frac{3}{2}} \tan{\left(\sqrt{x} \right)}} + \frac{\tan^{2}{\left(\sqrt{x} \right)} + 1}{x^{\frac{3}{2}} \tan{\left(\sqrt{x} \right)}}\right) \tan^{\log{\left(x \right)}}{\left(\sqrt{x} \right)}$$
The third derivative
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/ 3 / 2 \ \
|/ / / ___\\ / 2/ ___\\ \ / / / ___\\ / 2/ ___\\ \ | / / ___\\ / 2/ ___\\ / 2/ ___\\ / 2/ ___\\ / 2/ ___\\ | |
||2*log\tan\\/ x // \1 + tan \\/ x //*log(x)| |2*log\tan\\/ x // \1 + tan \\/ x //*log(x)| |4*log\tan\\/ x // 4*\1 + tan \\/ x // 2*\1 + tan \\/ x //*log(x) \1 + tan \\/ x // *log(x) \1 + tan \\/ x //*log(x)| |
||----------------- + ------------------------| 3*|----------------- + ------------------------|*|----------------- - ------------------- - -------------------------- + ------------------------- + ------------------------| 2 2 3 2 |
|| x ___ / ___\ | / / ___\\ | x ___ / ___\ | | 2 3/2 / ___\ x 2/ ___\ 3/2 / ___\ | / 2/ ___\\ / 2/ ___\\ / 2/ ___\\ / 2/ ___\\ / 2/ ___\\ / ___\ / 2/ ___\\ / 2/ ___\\ / 2/ ___\\ / 2/ ___\\ |
log(x)/ ___\ |\ \/ x *tan\\/ x / / 2*log\tan\\/ x // \ \/ x *tan\\/ x / / \ x x *tan\\/ x / x*tan \\/ x / x *tan\\/ x / / 3*\1 + tan \\/ x // 9*\1 + tan \\/ x // 3*\1 + tan \\/ x // 3*\1 + tan \\/ x //*log(x) \1 + tan \\/ x //*log(x)*tan\\/ x / \1 + tan \\/ x // *log(x) \1 + tan \\/ x // *log(x) 3*\1 + tan \\/ x // *log(x) 3*\1 + tan \\/ x //*log(x)|
tan \\/ x /*|----------------------------------------------- + ----------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + ------------------- - ------------------- - -------------------- - -------------------------- + ----------------------------------- - ------------------------- + ------------------------- + --------------------------- + --------------------------|
| 8 3 8 2 5/2 / ___\ 2 2/ ___\ 2 3/2 3/2 / ___\ 3/2 3/ ___\ 2 2/ ___\ 5/2 / ___\ |
\ x 2*x 4*x *tan\\/ x / 4*x *tan \\/ x / 4*x 2*x 2*x *tan\\/ x / 4*x *tan \\/ x / 8*x *tan \\/ x / 8*x *tan\\/ x / /
$$\left(\frac{\left(\frac{2 \log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{\sqrt{x} \tan{\left(\sqrt{x} \right)}}\right)^{3}}{8} - \frac{3 \left(\frac{2 \log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{\sqrt{x} \tan{\left(\sqrt{x} \right)}}\right) \left(\frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{2} \log{\left(x \right)}}{x \tan^{2}{\left(\sqrt{x} \right)}} - \frac{2 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{x} + \frac{4 \log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x^{2}} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{x^{\frac{3}{2}} \tan{\left(\sqrt{x} \right)}} - \frac{4 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)}{x^{\frac{3}{2}} \tan{\left(\sqrt{x} \right)}}\right)}{8} + \frac{3 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{2} \log{\left(x \right)}}{8 x^{2} \tan^{2}{\left(\sqrt{x} \right)}} - \frac{3 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{2}}{4 x^{2} \tan^{2}{\left(\sqrt{x} \right)}} - \frac{3 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{4 x^{2}} + \frac{3 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)}{2 x^{2}} + \frac{2 \log{\left(\tan{\left(\sqrt{x} \right)} \right)}}{x^{3}} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{3} \log{\left(x \right)}}{4 x^{\frac{3}{2}} \tan^{3}{\left(\sqrt{x} \right)}} - \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)^{2} \log{\left(x \right)}}{2 x^{\frac{3}{2}} \tan{\left(\sqrt{x} \right)}} + \frac{\left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)} \tan{\left(\sqrt{x} \right)}}{2 x^{\frac{3}{2}}} + \frac{3 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right) \log{\left(x \right)}}{8 x^{\frac{5}{2}} \tan{\left(\sqrt{x} \right)}} - \frac{9 \left(\tan^{2}{\left(\sqrt{x} \right)} + 1\right)}{4 x^{\frac{5}{2}} \tan{\left(\sqrt{x} \right)}}\right) \tan^{\log{\left(x \right)}}{\left(\sqrt{x} \right)}$$