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- Derivative of:
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Derivative of x^7*e^x
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Derivative of (x-1)/(x+1)
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Derivative of e^(x*(-3))
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Derivative of x/sin(x)
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Identical expressions
- arcsin(cos(4x))^ three
- arc sinus of ( co sinus of e of (4x)) cubed
- arc sinus of ( co sinus of e of (4x)) to the power of three
- arcsin(cos(4x))3
- arcsincos4x3
- arcsin(cos(4x))³
- arcsin(cos(4x)) to the power of 3
- arcsincos4x^3
Derivative of arcsin(cos(4x))^3
The solution
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3 asin (cos(4*x))
$$\operatorname{asin}^{3}{\left(\cos{\left(4 x \right)} \right)}$$
d / 3 \ --\asin (cos(4*x))/ dx
$$\frac{d}{d x} \operatorname{asin}^{3}{\left(\cos{\left(4 x \right)} \right)}$$
The first derivative
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2
-12*asin (cos(4*x))*sin(4*x)
----------------------------
_______________
/ 2
\/ 1 - cos (4*x)
$$- \frac{12 \sin{\left(4 x \right)} \operatorname{asin}^{2}{\left(\cos{\left(4 x \right)} \right)}}{\sqrt{- \cos^{2}{\left(4 x \right)} + 1}}$$
The second derivative
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/ 2 2 \ | 2*sin (4*x) asin(cos(4*x))*cos(4*x) sin (4*x)*asin(cos(4*x))*cos(4*x)| 48*|- -------------- - ----------------------- + ---------------------------------|*asin(cos(4*x)) | 2 _______________ 3/2 | | -1 + cos (4*x) / 2 / 2 \ | \ \/ 1 - cos (4*x) \1 - cos (4*x)/ /
$$48 \left(- \frac{2 \sin^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)} - 1} - \frac{\cos{\left(4 x \right)} \operatorname{asin}{\left(\cos{\left(4 x \right)} \right)}}{\sqrt{- \cos^{2}{\left(4 x \right)} + 1}} + \frac{\sin^{2}{\left(4 x \right)} \cos{\left(4 x \right)} \operatorname{asin}{\left(\cos{\left(4 x \right)} \right)}}{\left(- \cos^{2}{\left(4 x \right)} + 1\right)^{\frac{3}{2}}}\right) \operatorname{asin}{\left(\cos{\left(4 x \right)} \right)}$$
The third derivative
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/ 2 2 2 2 2 2 2 2 2 2 \
| asin (cos(4*x)) 2*sin (4*x) asin (cos(4*x))*sin (4*x) 6*asin(cos(4*x))*cos(4*x) 3*asin (cos(4*x))*cos (4*x) 6*sin (4*x)*asin(cos(4*x))*cos(4*x) 3*asin (cos(4*x))*cos (4*x)*sin (4*x)|
192*|------------------ - ------------------ - ------------------------- - ------------------------- + --------------------------- - ----------------------------------- - -------------------------------------|*sin(4*x)
| _______________ 3/2 3/2 2 3/2 2 5/2 |
| / 2 / 2 \ / 2 \ -1 + cos (4*x) / 2 \ / 2 \ / 2 \ |
\\/ 1 - cos (4*x) \1 - cos (4*x)/ \1 - cos (4*x)/ \1 - cos (4*x)/ \-1 + cos (4*x)/ \1 - cos (4*x)/ /
$$192 \left(- \frac{6 \sin^{2}{\left(4 x \right)} \cos{\left(4 x \right)} \operatorname{asin}{\left(\cos{\left(4 x \right)} \right)}}{\left(\cos^{2}{\left(4 x \right)} - 1\right)^{2}} - \frac{6 \cos{\left(4 x \right)} \operatorname{asin}{\left(\cos{\left(4 x \right)} \right)}}{\cos^{2}{\left(4 x \right)} - 1} + \frac{\operatorname{asin}^{2}{\left(\cos{\left(4 x \right)} \right)}}{\sqrt{- \cos^{2}{\left(4 x \right)} + 1}} - \frac{\sin^{2}{\left(4 x \right)} \operatorname{asin}^{2}{\left(\cos{\left(4 x \right)} \right)}}{\left(- \cos^{2}{\left(4 x \right)} + 1\right)^{\frac{3}{2}}} + \frac{3 \cos^{2}{\left(4 x \right)} \operatorname{asin}^{2}{\left(\cos{\left(4 x \right)} \right)}}{\left(- \cos^{2}{\left(4 x \right)} + 1\right)^{\frac{3}{2}}} - \frac{3 \sin^{2}{\left(4 x \right)} \cos^{2}{\left(4 x \right)} \operatorname{asin}^{2}{\left(\cos{\left(4 x \right)} \right)}}{\left(- \cos^{2}{\left(4 x \right)} + 1\right)^{\frac{5}{2}}} - \frac{2 \sin^{2}{\left(4 x \right)}}{\left(- \cos^{2}{\left(4 x \right)} + 1\right)^{\frac{3}{2}}}\right) \sin{\left(4 x \right)}$$
The graph