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- Derivative of a implicit defined function
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- Partial derivative of the function
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- Differential equations Step by Step
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How to use it?
- Derivative of:
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Derivative of f(x)=1
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Derivative of (x+8)^2
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Derivative of x^5-2*x
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Derivative of x^3-3
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Identical expressions
- (four *sin(six *x))^(five *x)
- (4 multiply by sinus of (6 multiply by x)) to the power of (5 multiply by x)
- (four multiply by sinus of (six multiply by x)) to the power of (five multiply by x)
- (4*sin(6*x))(5*x)
- 4*sin6*x5*x
- (4sin(6x))^(5x)
- (4sin(6x))(5x)
- 4sin6x5x
- 4sin6x^5x
Derivative of (4*sin(6*x))^(5*x)
The solution
You have entered
[src]
5*x (4*sin(6*x))
$$\left(4 \sin{\left(6 x \right)}\right)^{5 x}$$
(4*sin(6*x))^(5*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
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5*x / 30*x*cos(6*x)\
(4*sin(6*x)) *|5*log(4*sin(6*x)) + -------------|
\ sin(6*x) /
$$\left(4 \sin{\left(6 x \right)}\right)^{5 x} \left(\frac{30 x \cos{\left(6 x \right)}}{\sin{\left(6 x \right)}} + 5 \log{\left(4 \sin{\left(6 x \right)} \right)}\right)$$
The second derivative
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/ 2 2 \
5*x | /6*x*cos(6*x) \ 12*cos(6*x) 36*x*cos (6*x)|
5*(4*sin(6*x)) *|-36*x + 5*|------------ + log(4*sin(6*x))| + ----------- - --------------|
| \ sin(6*x) / sin(6*x) 2 |
\ sin (6*x) /
$$5 \left(4 \sin{\left(6 x \right)}\right)^{5 x} \left(- 36 x - \frac{36 x \cos^{2}{\left(6 x \right)}}{\sin^{2}{\left(6 x \right)}} + 5 \left(\frac{6 x \cos{\left(6 x \right)}}{\sin{\left(6 x \right)}} + \log{\left(4 \sin{\left(6 x \right)} \right)}\right)^{2} + \frac{12 \cos{\left(6 x \right)}}{\sin{\left(6 x \right)}}\right)$$
The third derivative
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/ 3 / 2 \ 2 3 \
5*x | /6*x*cos(6*x) \ /6*x*cos(6*x) \ | cos(6*x) 3*x*cos (6*x)| 108*cos (6*x) 432*x*cos (6*x) 432*x*cos(6*x)|
5*(4*sin(6*x)) *|-108 + 25*|------------ + log(4*sin(6*x))| - 180*|------------ + log(4*sin(6*x))|*|3*x - -------- + -------------| - ------------- + --------------- + --------------|
| \ sin(6*x) / \ sin(6*x) / | sin(6*x) 2 | 2 3 sin(6*x) |
\ \ sin (6*x) / sin (6*x) sin (6*x) /
$$5 \left(4 \sin{\left(6 x \right)}\right)^{5 x} \left(\frac{432 x \cos{\left(6 x \right)}}{\sin{\left(6 x \right)}} + \frac{432 x \cos^{3}{\left(6 x \right)}}{\sin^{3}{\left(6 x \right)}} + 25 \left(\frac{6 x \cos{\left(6 x \right)}}{\sin{\left(6 x \right)}} + \log{\left(4 \sin{\left(6 x \right)} \right)}\right)^{3} - 180 \left(\frac{6 x \cos{\left(6 x \right)}}{\sin{\left(6 x \right)}} + \log{\left(4 \sin{\left(6 x \right)} \right)}\right) \left(3 x + \frac{3 x \cos^{2}{\left(6 x \right)}}{\sin^{2}{\left(6 x \right)}} - \frac{\cos{\left(6 x \right)}}{\sin{\left(6 x \right)}}\right) - 108 - \frac{108 \cos^{2}{\left(6 x \right)}}{\sin^{2}{\left(6 x \right)}}\right)$$