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 sin(x)^2
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Derivative of x^-6
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Derivative of ex^2
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Derivative of e^sin(x)*cos(x)
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Identical expressions
- (log(x))^(cos(x))^ one / two
- ( logarithm of (x)) to the power of ( co sinus of e of (x)) to the power of 1 divide by 2
- ( logarithm of (x)) to the power of ( co sinus of e of (x)) to the power of one divide by two
- (log(x))(cos(x))1/2
- logxcosx1/2
- logx^cosx^1/2
- (log(x))^(cos(x))^1 divide by 2
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Similar expressions
- (log(x))^(cosx)^1/2
Derivative of (log(x))^(cos(x))^1/2
The solution
You have entered
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________
\/ cos(x)
(log(x))
$$\log{\left(x \right)}^{\sqrt{\cos{\left(x \right)}}}$$
log(x)^(sqrt(cos(x)))
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
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Now simplify:
The answer is:
The first derivative
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________ / ________ \
\/ cos(x) |\/ cos(x) log(log(x))*sin(x)|
(log(x)) *|---------- - ------------------|
| x*log(x) ________ |
\ 2*\/ cos(x) /
$$\left(- \frac{\log{\left(\log{\left(x \right)} \right)} \sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)}}} + \frac{\sqrt{\cos{\left(x \right)}}}{x \log{\left(x \right)}}\right) \log{\left(x \right)}^{\sqrt{\cos{\left(x \right)}}}$$
The second derivative
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/ 2 \
|/ ________\ |
||log(log(x))*sin(x) 2*\/ cos(x) | |
||------------------ - ------------| |
________ || ________ x*log(x) | ________ ________ ________ 2 |
\/ cos(x) |\ \/ cos(x) / \/ cos(x) *log(log(x)) \/ cos(x) \/ cos(x) sin (x)*log(log(x)) sin(x) |
(log(x)) *|------------------------------------ - ---------------------- - ---------- - ---------- - ------------------- - -------------------|
| 4 2 2 2 2 3/2 ________ |
\ x *log(x) x *log (x) 4*cos (x) x*\/ cos(x) *log(x)/
$$\left(\frac{\left(\frac{\log{\left(\log{\left(x \right)} \right)} \sin{\left(x \right)}}{\sqrt{\cos{\left(x \right)}}} - \frac{2 \sqrt{\cos{\left(x \right)}}}{x \log{\left(x \right)}}\right)^{2}}{4} - \frac{\log{\left(\log{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{4 \cos^{\frac{3}{2}}{\left(x \right)}} - \frac{\log{\left(\log{\left(x \right)} \right)} \sqrt{\cos{\left(x \right)}}}{2} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)} \sqrt{\cos{\left(x \right)}}} - \frac{\sqrt{\cos{\left(x \right)}}}{x^{2} \log{\left(x \right)}} - \frac{\sqrt{\cos{\left(x \right)}}}{x^{2} \log{\left(x \right)}^{2}}\right) \log{\left(x \right)}^{\sqrt{\cos{\left(x \right)}}}$$
The third derivative
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/ 3 \
| / ________\ / ________\ / 2 ________ ________ \ |
| |log(log(x))*sin(x) 2*\/ cos(x) | |log(log(x))*sin(x) 2*\/ cos(x) | | ________ sin (x)*log(log(x)) 4*\/ cos(x) 4*\/ cos(x) 4*sin(x) | |
| |------------------ - ------------| 3*|------------------ - ------------|*|2*\/ cos(x) *log(log(x)) + ------------------- + ------------ + ------------ + -------------------| |
________ | | ________ x*log(x) | | ________ x*log(x) | | 3/2 2 2 2 ________ | ________ ________ ________ ________ 3 2 |
\/ cos(x) | \ \/ cos(x) / \ \/ cos(x) / \ cos (x) x *log(x) x *log (x) x*\/ cos(x) *log(x)/ 2*\/ cos(x) 2*\/ cos(x) 3*\/ cos(x) 3*\/ cos(x) 3*sin (x)*log(log(x)) log(log(x))*sin(x) 3*sin (x) 3*sin(x) 3*sin(x) |
(log(x)) *|- ------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------ + ------------ + ------------ + ------------ - ------------ - --------------------- - ------------------ - -------------------- + ---------------------- + -----------------------|
| 8 8 3 3 3 3 2 2*x*log(x) 5/2 ________ 3/2 2 ________ 2 ________ 2 |
\ x *log(x) x *log (x) x *log (x) 8*cos (x) 4*\/ cos(x) 4*x*cos (x)*log(x) 2*x *\/ cos(x) *log(x) 2*x *\/ cos(x) *log (x)/
$$\left(- \frac{\left(\frac{\log{\left(\log{\left(x \right)} \right)} \sin{\left(x \right)}}{\sqrt{\cos{\left(x \right)}}} - \frac{2 \sqrt{\cos{\left(x \right)}}}{x \log{\left(x \right)}}\right)^{3}}{8} + \frac{3 \left(\frac{\log{\left(\log{\left(x \right)} \right)} \sin{\left(x \right)}}{\sqrt{\cos{\left(x \right)}}} - \frac{2 \sqrt{\cos{\left(x \right)}}}{x \log{\left(x \right)}}\right) \left(\frac{\log{\left(\log{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\cos^{\frac{3}{2}}{\left(x \right)}} + 2 \log{\left(\log{\left(x \right)} \right)} \sqrt{\cos{\left(x \right)}} + \frac{4 \sin{\left(x \right)}}{x \log{\left(x \right)} \sqrt{\cos{\left(x \right)}}} + \frac{4 \sqrt{\cos{\left(x \right)}}}{x^{2} \log{\left(x \right)}} + \frac{4 \sqrt{\cos{\left(x \right)}}}{x^{2} \log{\left(x \right)}^{2}}\right)}{8} - \frac{3 \log{\left(\log{\left(x \right)} \right)} \sin^{3}{\left(x \right)}}{8 \cos^{\frac{5}{2}}{\left(x \right)}} - \frac{\log{\left(\log{\left(x \right)} \right)} \sin{\left(x \right)}}{4 \sqrt{\cos{\left(x \right)}}} - \frac{3 \sin^{2}{\left(x \right)}}{4 x \log{\left(x \right)} \cos^{\frac{3}{2}}{\left(x \right)}} - \frac{3 \sqrt{\cos{\left(x \right)}}}{2 x \log{\left(x \right)}} + \frac{3 \sin{\left(x \right)}}{2 x^{2} \log{\left(x \right)} \sqrt{\cos{\left(x \right)}}} + \frac{3 \sin{\left(x \right)}}{2 x^{2} \log{\left(x \right)}^{2} \sqrt{\cos{\left(x \right)}}} + \frac{2 \sqrt{\cos{\left(x \right)}}}{x^{3} \log{\left(x \right)}} + \frac{3 \sqrt{\cos{\left(x \right)}}}{x^{3} \log{\left(x \right)}^{2}} + \frac{2 \sqrt{\cos{\left(x \right)}}}{x^{3} \log{\left(x \right)}^{3}}\right) \log{\left(x \right)}^{\sqrt{\cos{\left(x \right)}}}$$