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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 8^x
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Derivative of (x-2)^2*(x-4)+5
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Derivative of (x-1)/2
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Derivative of sin(x)*sin(x)
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Identical expressions
- (sinx)^(arccos(two *x))
- ( sinus of x) to the power of (arc co sinus of e of (2 multiply by x))
- ( sinus of x) to the power of (arc co sinus of e of (two multiply by x))
- (sinx)(arccos(2*x))
- sinxarccos2*x
- (sinx)^(arccos(2x))
- (sinx)(arccos(2x))
- sinxarccos2x
- sinx^arccos2x
Derivative of (sinx)^(arccos(2*x))
The solution
You have entered
[src]
acos(2*x) sin (x)
$$\sin^{\operatorname{acos}{\left(2 x \right)}}{\left(x \right)}$$
sin(x)^acos(2*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]
acos(2*x) / 2*log(sin(x)) acos(2*x)*cos(x)\
sin (x)*|- ------------- + ----------------|
| __________ sin(x) |
| / 2 |
\ \/ 1 - 4*x /
$$\left(\frac{\cos{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{1 - 4 x^{2}}}\right) \sin^{\operatorname{acos}{\left(2 x \right)}}{\left(x \right)}$$
The second derivative
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/ 2 2 \
acos(2*x) |/ 2*log(sin(x)) acos(2*x)*cos(x)\ cos (x)*acos(2*x) 8*x*log(sin(x)) 4*cos(x) |
sin (x)*||- ------------- + ----------------| - acos(2*x) - ----------------- - --------------- - --------------------|
|| __________ sin(x) | 2 3/2 __________ |
|| / 2 | sin (x) / 2\ / 2 |
\\ \/ 1 - 4*x / \1 - 4*x / \/ 1 - 4*x *sin(x)/
$$\left(- \frac{8 x \log{\left(\sin{\left(x \right)} \right)}}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}}} + \left(\frac{\cos{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{1 - 4 x^{2}}}\right)^{2} - \operatorname{acos}{\left(2 x \right)} - \frac{\cos^{2}{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin^{2}{\left(x \right)}} - \frac{4 \cos{\left(x \right)}}{\sqrt{1 - 4 x^{2}} \sin{\left(x \right)}}\right) \sin^{\operatorname{acos}{\left(2 x \right)}}{\left(x \right)}$$
The third derivative
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/ 3 / 2 \ 2 3 2 \
acos(2*x) |/ 2*log(sin(x)) acos(2*x)*cos(x)\ 6 8*log(sin(x)) / 2*log(sin(x)) acos(2*x)*cos(x)\ |cos (x)*acos(2*x) 4*cos(x) 8*x*log(sin(x)) | 96*x *log(sin(x)) 2*cos (x)*acos(2*x) 2*acos(2*x)*cos(x) 6*cos (x) 24*x*cos(x) |
sin (x)*||- ------------- + ----------------| + ------------- - ------------- - 3*|- ------------- + ----------------|*|----------------- + -------------------- + --------------- + acos(2*x)| - ----------------- + ------------------- + ------------------ + --------------------- - --------------------|
|| __________ sin(x) | __________ 3/2 | __________ sin(x) | | 2 __________ 3/2 | 5/2 3 sin(x) __________ 3/2 |
|| / 2 | / 2 / 2\ | / 2 | | sin (x) / 2 / 2\ | / 2\ sin (x) / 2 2 / 2\ |
\\ \/ 1 - 4*x / \/ 1 - 4*x \1 - 4*x / \ \/ 1 - 4*x / \ \/ 1 - 4*x *sin(x) \1 - 4*x / / \1 - 4*x / \/ 1 - 4*x *sin (x) \1 - 4*x / *sin(x)/
$$\left(- \frac{96 x^{2} \log{\left(\sin{\left(x \right)} \right)}}{\left(1 - 4 x^{2}\right)^{\frac{5}{2}}} - \frac{24 x \cos{\left(x \right)}}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}} \sin{\left(x \right)}} + \left(\frac{\cos{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{1 - 4 x^{2}}}\right)^{3} - 3 \left(\frac{\cos{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{1 - 4 x^{2}}}\right) \left(\frac{8 x \log{\left(\sin{\left(x \right)} \right)}}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}}} + \operatorname{acos}{\left(2 x \right)} + \frac{\cos^{2}{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin^{2}{\left(x \right)}} + \frac{4 \cos{\left(x \right)}}{\sqrt{1 - 4 x^{2}} \sin{\left(x \right)}}\right) + \frac{2 \cos{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin{\left(x \right)}} + \frac{2 \cos^{3}{\left(x \right)} \operatorname{acos}{\left(2 x \right)}}{\sin^{3}{\left(x \right)}} + \frac{6}{\sqrt{1 - 4 x^{2}}} + \frac{6 \cos^{2}{\left(x \right)}}{\sqrt{1 - 4 x^{2}} \sin^{2}{\left(x \right)}} - \frac{8 \log{\left(\sin{\left(x \right)} \right)}}{\left(1 - 4 x^{2}\right)^{\frac{3}{2}}}\right) \sin^{\operatorname{acos}{\left(2 x \right)}}{\left(x \right)}$$