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How to use it?
- Derivative of:
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Derivative of (-2)/x^2
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Derivative of (cos(x))^sin(x)
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Derivative of y=(x+2)²(2x-1)³
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Derivative of y=5x-6
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Identical expressions
- y=(log5x)^arccosx
- y equally ( logarithm of 5x) to the power of arc co sinus of e of x
- y=(log5x)arccosx
- y=log5xarccosx
- y=log5x^arccosx
Derivative of y=(log5x)^arccosx
The solution
You have entered
[src]
acos(x) log (5*x)
$$\log{\left(5 x \right)}^{\operatorname{acos}{\left(x \right)}}$$
log(5*x)^acos(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
[src]
acos(x) / log(log(5*x)) acos(x) \
log (5*x)*|- ------------- + ----------|
| ________ x*log(5*x)|
| / 2 |
\ \/ 1 - x /
$$\left(- \frac{\log{\left(\log{\left(5 x \right)} \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{acos}{\left(x \right)}}{x \log{\left(5 x \right)}}\right) \log{\left(5 x \right)}^{\operatorname{acos}{\left(x \right)}}$$
The second derivative
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/ 2 \
acos(x) |/log(log(5*x)) acos(x) \ x*log(log(5*x)) acos(x) acos(x) 2 |
log (5*x)*||------------- - ----------| - --------------- - ----------- - ------------ - ----------------------|
|| ________ x*log(5*x)| 3/2 2 2 2 ________ |
|| / 2 | / 2\ x *log(5*x) x *log (5*x) / 2 |
\\ \/ 1 - x / \1 - x / x*\/ 1 - x *log(5*x)/
$$\left(- \frac{x \log{\left(\log{\left(5 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \left(\frac{\log{\left(\log{\left(5 x \right)} \right)}}{\sqrt{1 - x^{2}}} - \frac{\operatorname{acos}{\left(x \right)}}{x \log{\left(5 x \right)}}\right)^{2} - \frac{2}{x \sqrt{1 - x^{2}} \log{\left(5 x \right)}} - \frac{\operatorname{acos}{\left(x \right)}}{x^{2} \log{\left(5 x \right)}} - \frac{\operatorname{acos}{\left(x \right)}}{x^{2} \log{\left(5 x \right)}^{2}}\right) \log{\left(5 x \right)}^{\operatorname{acos}{\left(x \right)}}$$
The third derivative
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/ 3 2 \
acos(x) | /log(log(5*x)) acos(x) \ log(log(5*x)) 3 /log(log(5*x)) acos(x) \ /x*log(log(5*x)) acos(x) acos(x) 2 \ 3*x *log(log(5*x)) 2*acos(x) 2*acos(x) 3*acos(x) 3 3 |
log (5*x)*|- |------------- - ----------| - ------------- - -------------------- + 3*|------------- - ----------|*|--------------- + ----------- + ------------ + ----------------------| - ------------------ + ----------- + ------------ + ------------ + ----------------------- + ------------------------|
| | ________ x*log(5*x)| 3/2 3/2 | ________ x*log(5*x)| | 3/2 2 2 2 ________ | 5/2 3 3 3 3 2 ________ ________ |
| | / 2 | / 2\ / 2\ | / 2 | | / 2\ x *log(5*x) x *log (5*x) / 2 | / 2\ x *log(5*x) x *log (5*x) x *log (5*x) 2 / 2 2 / 2 2 |
\ \ \/ 1 - x / \1 - x / \1 - x / *log(5*x) \ \/ 1 - x / \ \1 - x / x*\/ 1 - x *log(5*x)/ \1 - x / x *\/ 1 - x *log(5*x) x *\/ 1 - x *log (5*x)/
$$\left(- \frac{3 x^{2} \log{\left(\log{\left(5 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} - \left(\frac{\log{\left(\log{\left(5 x \right)} \right)}}{\sqrt{1 - x^{2}}} - \frac{\operatorname{acos}{\left(x \right)}}{x \log{\left(5 x \right)}}\right)^{3} + 3 \left(\frac{\log{\left(\log{\left(5 x \right)} \right)}}{\sqrt{1 - x^{2}}} - \frac{\operatorname{acos}{\left(x \right)}}{x \log{\left(5 x \right)}}\right) \left(\frac{x \log{\left(\log{\left(5 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{2}{x \sqrt{1 - x^{2}} \log{\left(5 x \right)}} + \frac{\operatorname{acos}{\left(x \right)}}{x^{2} \log{\left(5 x \right)}} + \frac{\operatorname{acos}{\left(x \right)}}{x^{2} \log{\left(5 x \right)}^{2}}\right) - \frac{\log{\left(\log{\left(5 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{3}{\left(1 - x^{2}\right)^{\frac{3}{2}} \log{\left(5 x \right)}} + \frac{3}{x^{2} \sqrt{1 - x^{2}} \log{\left(5 x \right)}} + \frac{3}{x^{2} \sqrt{1 - x^{2}} \log{\left(5 x \right)}^{2}} + \frac{2 \operatorname{acos}{\left(x \right)}}{x^{3} \log{\left(5 x \right)}} + \frac{3 \operatorname{acos}{\left(x \right)}}{x^{3} \log{\left(5 x \right)}^{2}} + \frac{2 \operatorname{acos}{\left(x \right)}}{x^{3} \log{\left(5 x \right)}^{3}}\right) \log{\left(5 x \right)}^{\operatorname{acos}{\left(x \right)}}$$
The graph