Mister Exam
Other calculators
- Derivative of a implicit defined function
- Derivative of Parametric Function
- Partial derivative of the function
- Curve tracing functions Step by Step
- Integral Step by Step
- Differential equations Step by Step
- Limits Step by Step
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How to use it?
- Derivative of:
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Derivative of x^3/cosx
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Derivative of (x-6)x^3
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Derivative of y=x(x-1)
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Derivative of y=5+12x-x³
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Identical expressions
- (cosx)^loge(x)
- ( co sinus of e of x) to the power of logarithm of e(x)
- (cosx)loge(x)
- cosxlogex
- cosx^logex
Derivative of (cosx)^loge(x)
The solution
You have entered
[src]
log(x)
cos (x)
------------
/ 1\
log\e /
$$\frac{\cos^{\log{\left(x \right)}}{\left(x \right)}}{\log{\left(e^{1} \right)}}$$
cos(x)^log(x)/log(exp(1))
Detail solution
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The derivative of a constant times a function is the constant times the derivative of the function.
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Don't know the steps in finding this derivative.
But the derivative is
So, the result is:
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Now simplify:
The answer is:
The first derivative
[src]
log(x) /log(cos(x)) log(x)*sin(x)\
cos (x)*|----------- - -------------|
\ x cos(x) /
------------------------------------------
/ 1\
log\e /
$$\frac{\left(- \frac{\log{\left(x \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\right) \cos^{\log{\left(x \right)}}{\left(x \right)}}{\log{\left(e^{1} \right)}}$$
The second derivative
[src]
/ 2 2 \
log(x) | / log(cos(x)) log(x)*sin(x)\ log(cos(x)) sin (x)*log(x) 2*sin(x) |
-cos (x)*|- |- ----------- + -------------| + ----------- + -------------- + -------- + log(x)|
| \ x cos(x) / 2 2 x*cos(x) |
\ x cos (x) /
------------------------------------------------------------------------------------------------------
/ 1\
log\e /
$$- \frac{\left(- \left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\right)^{2} + \frac{\log{\left(x \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \log{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \cos^{\log{\left(x \right)}}{\left(x \right)}}{\log{\left(e^{1} \right)}}$$
The third derivative
[src]
/ 3 / 2 \ 3 2 \
log(x) |/ log(cos(x)) log(x)*sin(x)\ 3 / log(cos(x)) log(x)*sin(x)\ |log(cos(x)) sin (x)*log(x) 2*sin(x) | 2*log(cos(x)) 3*sin(x) 2*log(x)*sin(x) 2*sin (x)*log(x) 3*sin (x)|
-cos (x)*||- ----------- + -------------| + - - 3*|- ----------- + -------------|*|----------- + -------------- + -------- + log(x)| - ------------- - --------- + --------------- + ---------------- + ---------|
|\ x cos(x) / x \ x cos(x) / | 2 2 x*cos(x) | 3 2 cos(x) 3 2 |
\ \ x cos (x) / x x *cos(x) cos (x) x*cos (x)/
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
/ 1\
log\e /
$$- \frac{\left(\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\right)^{3} - 3 \left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\right) \left(\frac{\log{\left(x \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \log{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) + \frac{2 \log{\left(x \right)} \sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{2 \log{\left(x \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \sin^{2}{\left(x \right)}}{x \cos^{2}{\left(x \right)}} + \frac{3}{x} - \frac{3 \sin{\left(x \right)}}{x^{2} \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{3}}\right) \cos^{\log{\left(x \right)}}{\left(x \right)}}{\log{\left(e^{1} \right)}}$$