Mister Exam
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- Derivative of a implicit defined function
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- Partial derivative of the function
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How to use it?
- Derivative of:
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Derivative of x^2*sqrt(x)
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Derivative of x*(log(x)/log(10))
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Derivative of x/5
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Derivative of x^2+x+1
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Identical expressions
- y=(one +cosx)^x^ five
- y equally (1 plus co sinus of e of x) to the power of x to the power of 5
- y equally (one plus co sinus of e of x) to the power of x to the power of five
- y=(1+cosx)x5
- y=1+cosxx5
- y=(1+cosx)^x⁵
- y=1+cosx^x^5
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Similar expressions
- y=(1-cosx)^x^5
Derivative of y=(1+cosx)^x^5
The solution
You have entered
[src]
/ 5\
\x /
(1 + cos(x))
$$\left(\cos{\left(x \right)} + 1\right)^{x^{5}}$$
/ / 5\\ d | \x /| --\(1 + cos(x)) / dx
$$\frac{d}{d x} \left(\cos{\left(x \right)} + 1\right)^{x^{5}}$$
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]
/ 5\ / 5 \
\x / | 4 x *sin(x) |
(1 + cos(x)) *|5*x *log(1 + cos(x)) - ----------|
\ 1 + cos(x)/
$$\left(- \frac{x^{5} \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} + 5 x^{4} \log{\left(\cos{\left(x \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} + 1\right)^{x^{5}}$$
The second derivative
[src]
/ 5\ / 2 2 2 2 \
3 \x / | 5 / x*sin(x) \ x *cos(x) x *sin (x) 10*x*sin(x)|
x *(1 + cos(x)) *|20*log(1 + cos(x)) + x *|-5*log(1 + cos(x)) + ----------| - ---------- - ------------- - -----------|
| \ 1 + cos(x)/ 1 + cos(x) 2 1 + cos(x)|
\ (1 + cos(x)) /
$$x^{3} \left(\cos{\left(x \right)} + 1\right)^{x^{5}} \left(x^{5} \left(\frac{x \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} - 5 \log{\left(\cos{\left(x \right)} + 1 \right)}\right)^{2} - \frac{x^{2} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} - \frac{x^{2} \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} - \frac{10 x \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} + 20 \log{\left(\cos{\left(x \right)} + 1 \right)}\right)$$
The third derivative
[src]
/ 5\ / 3 3 2 2 2 3 3 / 2 2 2 \ 3 \
2 \x / | 10 / x*sin(x) \ x *sin(x) 60*x*sin(x) 15*x *cos(x) 15*x *sin (x) 2*x *sin (x) 5 / x*sin(x) \ | x *cos(x) x *sin (x) 10*x*sin(x)| 3*x *cos(x)*sin(x)|
x *(1 + cos(x)) *|60*log(1 + cos(x)) - x *|-5*log(1 + cos(x)) + ----------| + ---------- - ----------- - ------------ - ------------- - ------------- + 3*x *|-5*log(1 + cos(x)) + ----------|*|-20*log(1 + cos(x)) + ---------- + ------------- + -----------| - ------------------|
| \ 1 + cos(x)/ 1 + cos(x) 1 + cos(x) 1 + cos(x) 2 3 \ 1 + cos(x)/ | 1 + cos(x) 2 1 + cos(x)| 2 |
\ (1 + cos(x)) (1 + cos(x)) \ (1 + cos(x)) / (1 + cos(x)) /
$$x^{2} \left(\cos{\left(x \right)} + 1\right)^{x^{5}} \left(- x^{10} \left(\frac{x \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} - 5 \log{\left(\cos{\left(x \right)} + 1 \right)}\right)^{3} + 3 x^{5} \left(\frac{x \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} - 5 \log{\left(\cos{\left(x \right)} + 1 \right)}\right) \left(\frac{x^{2} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{x^{2} \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \frac{10 x \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} - 20 \log{\left(\cos{\left(x \right)} + 1 \right)}\right) + \frac{x^{3} \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} - \frac{3 x^{3} \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} - \frac{2 x^{3} \sin^{3}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{3}} - \frac{15 x^{2} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} - \frac{15 x^{2} \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} - \frac{60 x \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} + 60 \log{\left(\cos{\left(x \right)} + 1 \right)}\right)$$
The graph