Detail solution
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Let .
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The derivative of is .
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Then, apply the chain rule. Multiply by :
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Apply the quotient rule, which is:
and .
To find :
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Apply the product rule:
; to find :
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; to find :
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The derivative of cosine is negative sine:
; to find :
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The derivative of sine is cosine:
The result is:
To find :
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The derivative of is itself.
Now plug in to the quotient rule:
The result of the chain rule is:
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Now simplify:
The answer is:
The first derivative
[src]
-x / x 2 -x x 2 -x x -x x -x \ x
2 *\2 *cos (x)*e - 2 *sin (x)*e - 2 *cos(x)*e *sin(x) + 2 *cos(x)*e *log(2)*sin(x)/*e
---------------------------------------------------------------------------------------------
cos(x)*sin(x)
$$\frac{2^{- x} \left(- 2^{x} e^{- x} \sin^{2}{\left(x \right)} - 2^{x} e^{- x} \sin{\left(x \right)} \cos{\left(x \right)} + 2^{x} e^{- x} \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 2^{x} e^{- x} \cos^{2}{\left(x \right)}\right) e^{x}}{\sin{\left(x \right)} \cos{\left(x \right)}}$$
The second derivative
[src]
2 2 2 2 2 2 2 2 2 2 2 / 2 2 \
sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x) sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x) sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x) - 2*sin (x) + 2*cos (x) - 2*cos (x)*log(2) + 2*sin (x)*log(2) + 3*cos(x)*sin(x) - log (2)*cos(x)*sin(x) + 2*cos(x)*log(2)*sin(x) \sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2)
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2 2 cos(x)*sin(x) cos(x)*sin(x) cos(x)*sin(x)
sin (x) cos (x)
$$- \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{\left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{- 2 \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}$$
The third derivative
[src]
/ 2 2 \ / 2 2 2 2 2 \ / 2 2 \ / 2 2 2 2 2 \ 2 2 2 2 2 2 2 2 3 2 2 2 / 2 2 2 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ 2 / 2 2 \ / 2 2 \ / 2 2 2 2 2 \
2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/ 2*\- 2*sin (x) + 2*cos (x) - 2*cos (x)*log(2) + 2*sin (x)*log(2) + 3*cos(x)*sin(x) - log (2)*cos(x)*sin(x) + 2*cos(x)*log(2)*sin(x)/ 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/ 2*\- 2*sin (x) + 2*cos (x) - 2*cos (x)*log(2) + 2*sin (x)*log(2) + 3*cos(x)*sin(x) - log (2)*cos(x)*sin(x) + 2*cos(x)*log(2)*sin(x)/ sin (x) - cos (x) - 6*cos (x)*log(2) - 3*log (2)*sin (x) + 3*cos (x)*log (2) + 6*sin (x)*log(2) + 11*cos(x)*sin(x) + log (2)*cos(x)*sin(x) - 9*cos(x)*log(2)*sin(x) - 3*log (2)*cos(x)*sin(x) sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x) 2*\- 2*sin (x) + 2*cos (x) - 2*cos (x)*log(2) + 2*sin (x)*log(2) + 3*cos(x)*sin(x) - log (2)*cos(x)*sin(x) + 2*cos(x)*log(2)*sin(x)/ 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*sin(x) 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*cos(x) 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2) 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2) log (2)*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/ 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2) 2*\- 2*sin (x) + 2*cos (x) - 2*cos (x)*log(2) + 2*sin (x)*log(2) + 3*cos(x)*sin(x) - log (2)*cos(x)*sin(x) + 2*cos(x)*log(2)*sin(x)/*log(2)
- ------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------ - ------------------------------------------------------------------- - ------------------------------------------------------------------- - ------------------------------------------------------------------- + ------------------------------------------------------------------- - ------------------------------------------------------------------ + ------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------
2 2 2 2 cos(x)*sin(x) cos(x)*sin(x) cos(x)*sin(x) 3 3 2 2 cos(x)*sin(x) cos(x)*sin(x) cos(x)*sin(x)
cos (x) cos (x) sin (x) sin (x) cos (x) sin (x) sin (x) cos (x)
$$- \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}^{2}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{2 \left(- 2 \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \frac{2 \left(- 2 \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(- 2 \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(- 2 \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)}} + \frac{- 3 \log{\left(2 \right)}^{2} \sin^{2}{\left(x \right)} + \sin^{2}{\left(x \right)} + 6 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - 9 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} - 3 \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)}^{3} \sin{\left(x \right)} \cos{\left(x \right)} + 11 \sin{\left(x \right)} \cos{\left(x \right)} - 6 \log{\left(2 \right)} \cos^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + 3 \log{\left(2 \right)}^{2} \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}$$