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- Derivative of a implicit defined function
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How to use it?
- Derivative of:
- Derivative of |x|/x
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Derivative of ln(1/x)
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Derivative of ln(1+x)
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Derivative of sin(cosx)
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Identical expressions
- sin^nx*cosnx
- sinus of to the power of nx multiply by co sinus of e of nx
- sinnx*cosnx
- sin^nxcosnx
- sinnxcosnx
Derivative of sin^nx*cosnx
The solution
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n*x sin (cos(n*x))
$$\sin^{n x}{\left(\cos{\left(n x \right)} \right)}$$
sin(cos(n*x))^(n*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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/ 2 \
n*x | x*n *cos(cos(n*x))*sin(n*x)|
sin (cos(n*x))*|n*log(sin(cos(n*x))) - ---------------------------|
\ sin(cos(n*x)) /
$$\left(- \frac{n^{2} x \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} + n \log{\left(\sin{\left(\cos{\left(n x \right)} \right)} \right)}\right) \sin^{n x}{\left(\cos{\left(n x \right)} \right)}$$
The second derivative
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/ 2 2 2 \
2 n*x |/ n*x*cos(cos(n*x))*sin(n*x)\ 2 2*cos(cos(n*x))*sin(n*x) n*x*cos (cos(n*x))*sin (n*x) n*x*cos(n*x)*cos(cos(n*x))|
n *sin (cos(n*x))*||-log(sin(cos(n*x))) + --------------------------| - n*x*sin (n*x) - ------------------------ - ---------------------------- - --------------------------|
|\ sin(cos(n*x)) / sin(cos(n*x)) 2 sin(cos(n*x)) |
\ sin (cos(n*x)) /
$$n^{2} \left(- n x \sin^{2}{\left(n x \right)} - \frac{n x \sin^{2}{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}} - \frac{n x \cos{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} + \left(\frac{n x \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \log{\left(\sin{\left(\cos{\left(n x \right)} \right)} \right)}\right)^{2} - \frac{2 \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}}\right) \sin^{n x}{\left(\cos{\left(n x \right)} \right)}$$
The third derivative
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/ 3 / 2 2 \ 2 2 3 3 3 2 \
3 n*x | / n*x*cos(cos(n*x))*sin(n*x)\ 2 / n*x*cos(cos(n*x))*sin(n*x)\ | 2 2*cos(cos(n*x))*sin(n*x) n*x*cos (cos(n*x))*sin (n*x) n*x*cos(n*x)*cos(cos(n*x))| 3*cos (cos(n*x))*sin (n*x) 3*cos(n*x)*cos(cos(n*x)) n*x*cos(cos(n*x))*sin(n*x) 2*n*x*cos (cos(n*x))*sin (n*x) 2*n*x*sin (n*x)*cos(cos(n*x)) 3*n*x*cos (cos(n*x))*cos(n*x)*sin(n*x)|
n *sin (cos(n*x))*|- |-log(sin(cos(n*x))) + --------------------------| - 3*sin (n*x) + 3*|-log(sin(cos(n*x))) + --------------------------|*|n*x*sin (n*x) + ------------------------ + ---------------------------- + --------------------------| - -------------------------- - ------------------------ - 3*n*x*cos(n*x)*sin(n*x) + -------------------------- - ------------------------------ - ----------------------------- - --------------------------------------|
| \ sin(cos(n*x)) / \ sin(cos(n*x)) / | sin(cos(n*x)) 2 sin(cos(n*x)) | 2 sin(cos(n*x)) sin(cos(n*x)) 3 sin(cos(n*x)) 2 |
\ \ sin (cos(n*x)) / sin (cos(n*x)) sin (cos(n*x)) sin (cos(n*x)) /
$$n^{3} \left(- \frac{2 n x \sin^{3}{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \frac{2 n x \sin^{3}{\left(n x \right)} \cos^{3}{\left(\cos{\left(n x \right)} \right)}}{\sin^{3}{\left(\cos{\left(n x \right)} \right)}} - 3 n x \sin{\left(n x \right)} \cos{\left(n x \right)} + \frac{n x \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \frac{3 n x \sin{\left(n x \right)} \cos{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}} - \left(\frac{n x \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \log{\left(\sin{\left(\cos{\left(n x \right)} \right)} \right)}\right)^{3} + 3 \left(\frac{n x \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \log{\left(\sin{\left(\cos{\left(n x \right)} \right)} \right)}\right) \left(n x \sin^{2}{\left(n x \right)} + \frac{n x \sin^{2}{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}} + \frac{n x \cos{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} + \frac{2 \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}}\right) - 3 \sin^{2}{\left(n x \right)} - \frac{3 \sin^{2}{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}} - \frac{3 \cos{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}}\right) \sin^{n x}{\left(\cos{\left(n x \right)} \right)}$$