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- Derivative of a implicit defined function
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How to use it?
- Derivative of:
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Derivative of 4/x^2
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Derivative of cos(x^3)
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Derivative of (cos(x))^3
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Derivative of 1/tan(x)
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Identical expressions
- (sinx)^(ln(sinx)/ five)
- ( sinus of x) to the power of (ln( sinus of x) divide by 5)
- ( sinus of x) to the power of (ln( sinus of x) divide by five)
- (sinx)(ln(sinx)/5)
- sinxlnsinx/5
- sinx^lnsinx/5
- (sinx)^(ln(sinx) divide by 5)
Derivative of (sinx)^(ln(sinx)/5)
The solution
You have entered
[src]
log(sin(x))
-----------
5
(sin(x))
$$\sin^{\frac{\log{\left(\sin{\left(x \right)} \right)}}{5}}{\left(x \right)}$$
sin(x)^(log(sin(x))/5)
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
[src]
log(sin(x))
-----------
5
2*(sin(x)) *cos(x)*log(sin(x))
----------------------------------------
5*sin(x)
$$\frac{2 \log{\left(\sin{\left(x \right)} \right)} \sin^{\frac{\log{\left(\sin{\left(x \right)} \right)}}{5}}{\left(x \right)} \cos{\left(x \right)}}{5 \sin{\left(x \right)}}$$
The second derivative
[src]
log(sin(x))
----------- / 2 2 2 2 \
5 | 5*cos (x) 5*cos (x)*log(sin(x)) 2*cos (x)*log (sin(x))|
2*(sin(x)) *|-5*log(sin(x)) + --------- - --------------------- + ----------------------|
| 2 2 2 |
\ sin (x) sin (x) sin (x) /
---------------------------------------------------------------------------------------------------
25
$$\frac{2 \left(\frac{2 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 5 \log{\left(\sin{\left(x \right)} \right)} - \frac{5 \log{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{5 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{\frac{\log{\left(\sin{\left(x \right)} \right)}}{5}}{\left(x \right)}}{25}$$
The third derivative
[src]
log(sin(x))
----------- / 2 2 2 2 3 2 \
5 | 2 75*cos (x) 30*cos (x)*log (sin(x)) 4*cos (x)*log (sin(x)) 80*cos (x)*log(sin(x))|
2*(sin(x)) *|-75 - 30*log (sin(x)) + 50*log(sin(x)) - ---------- - ----------------------- + ---------------------- + ----------------------|*cos(x)
| 2 2 2 2 |
\ sin (x) sin (x) sin (x) sin (x) /
--------------------------------------------------------------------------------------------------------------------------------------------------------------
125*sin(x)
$$\frac{2 \left(\frac{4 \log{\left(\sin{\left(x \right)} \right)}^{3} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 30 \log{\left(\sin{\left(x \right)} \right)}^{2} - \frac{30 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 50 \log{\left(\sin{\left(x \right)} \right)} + \frac{80 \log{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 75 - \frac{75 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{\frac{\log{\left(\sin{\left(x \right)} \right)}}{5}}{\left(x \right)} \cos{\left(x \right)}}{125 \sin{\left(x \right)}}$$