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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 9/x
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Derivative of x^-5
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Derivative of x^2*e^x
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Derivative of x^2*cos(x)
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Identical expressions
- (log2sinx)^(4x)
- ( logarithm of 2 sinus of x) to the power of (4x)
- (log2sinx)(4x)
- log2sinx4x
- log2sinx^4x
Derivative of (log2sinx)^(4x)
The solution
You have entered
[src]
4*x log (2*sin(x))
$$\log{\left(2 \sin{\left(x \right)} \right)}^{4 x}$$
log(2*sin(x))^(4*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]
4*x / 4*x*cos(x) \
log (2*sin(x))*|4*log(log(2*sin(x))) + --------------------|
\ log(2*sin(x))*sin(x)/
$$\left(\frac{4 x \cos{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin{\left(x \right)}} + 4 \log{\left(\log{\left(2 \sin{\left(x \right)} \right)} \right)}\right) \log{\left(2 \sin{\left(x \right)} \right)}^{4 x}$$
The second derivative
[src]
/ 2 2 \
| 2*cos(x) x*cos (x) x*cos (x) |
| x - -------- + --------- + ---------------------|
| 2 sin(x) 2 2 |
4*x | / x*cos(x) \ sin (x) log(2*sin(x))*sin (x)|
4*log (2*sin(x))*|4*|-------------------- + log(log(2*sin(x)))| - ------------------------------------------------|
\ \log(2*sin(x))*sin(x) / log(2*sin(x)) /
$$4 \left(4 \left(\frac{x \cos{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin{\left(x \right)}} + \log{\left(\log{\left(2 \sin{\left(x \right)} \right)} \right)}\right)^{2} - \frac{x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{x \cos^{2}{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{\log{\left(2 \sin{\left(x \right)} \right)}}\right) \log{\left(2 \sin{\left(x \right)} \right)}^{4 x}$$
The third derivative
[src]
/ 2 2 3 3 3 / 2 2 \\
| 3*cos (x) 3*cos (x) 2*x*cos (x) 2*x*cos(x) 2*x*cos (x) 3*x*cos (x) 3*x*cos(x) / x*cos(x) \ | 2*cos(x) x*cos (x) x*cos (x) ||
| -3 - --------- - --------------------- + ----------- + ---------- + ---------------------- + --------------------- + -------------------- 12*|-------------------- + log(log(2*sin(x)))|*|x - -------- + --------- + ---------------------||
| 3 2 2 3 sin(x) 2 3 3 log(2*sin(x))*sin(x) \log(2*sin(x))*sin(x) / | sin(x) 2 2 ||
4*x | / x*cos(x) \ sin (x) log(2*sin(x))*sin (x) sin (x) log (2*sin(x))*sin (x) log(2*sin(x))*sin (x) \ sin (x) log(2*sin(x))*sin (x)/|
4*log (2*sin(x))*|16*|-------------------- + log(log(2*sin(x)))| + ----------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------|
\ \log(2*sin(x))*sin(x) / log(2*sin(x)) log(2*sin(x)) /
$$4 \left(16 \left(\frac{x \cos{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin{\left(x \right)}} + \log{\left(\log{\left(2 \sin{\left(x \right)} \right)} \right)}\right)^{3} - \frac{12 \left(\frac{x \cos{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin{\left(x \right)}} + \log{\left(\log{\left(2 \sin{\left(x \right)} \right)} \right)}\right) \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{x \cos^{2}{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)}{\log{\left(2 \sin{\left(x \right)} \right)}} + \frac{\frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 x \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{3 x \cos{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin{\left(x \right)}} + \frac{3 x \cos^{3}{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin^{3}{\left(x \right)}} + \frac{2 x \cos^{3}{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)}^{2} \sin^{3}{\left(x \right)}} - 3 - \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{3 \cos^{2}{\left(x \right)}}{\log{\left(2 \sin{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}}{\log{\left(2 \sin{\left(x \right)} \right)}}\right) \log{\left(2 \sin{\left(x \right)} \right)}^{4 x}$$