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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 2*sinh(x)*cosh(x)
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Derivative of x^2*asinh(x)*acosh(x)
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Derivative of (x^2-4)/x
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Identical expressions
- (sin(three *x))^acos(x)
- ( sinus of (3 multiply by x)) to the power of arc co sinus of e of ine of (x)
- ( sinus of (three multiply by x)) to the power of arc co sinus of e of ine of (x)
- (sin(3*x))acos(x)
- sin3*xacosx
- (sin(3x))^acos(x)
- (sin(3x))acos(x)
- sin3xacosx
- sin3x^acosx
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Similar expressions
- (sin(3*x))^arccos(x)
- (sin(3*x))^arccosx
Derivative of (sin(3*x))^acos(x)
The solution
You have entered
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acos(x) sin (3*x)
$$\sin^{\operatorname{acos}{\left(x \right)}}{\left(3 x \right)}$$
sin(3*x)^acos(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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acos(x) / log(sin(3*x)) 3*acos(x)*cos(3*x)\
sin (3*x)*|- ------------- + ------------------|
| ________ sin(3*x) |
| / 2 |
\ \/ 1 - x /
$$\left(\frac{3 \cos{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{1 - x^{2}}}\right) \sin^{\operatorname{acos}{\left(x \right)}}{\left(3 x \right)}$$
The second derivative
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/ 2 2 \
acos(x) |/ log(sin(3*x)) 3*acos(x)*cos(3*x)\ x*log(sin(3*x)) 9*cos (3*x)*acos(x) 6*cos(3*x) |
sin (3*x)*||- ------------- + ------------------| - 9*acos(x) - --------------- - ------------------- - --------------------|
|| ________ sin(3*x) | 3/2 2 ________ |
|| / 2 | / 2\ sin (3*x) / 2 |
\\ \/ 1 - x / \1 - x / \/ 1 - x *sin(3*x)/
$$\left(- \frac{x \log{\left(\sin{\left(3 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \left(\frac{3 \cos{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{1 - x^{2}}}\right)^{2} - 9 \operatorname{acos}{\left(x \right)} - \frac{9 \cos^{2}{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin^{2}{\left(3 x \right)}} - \frac{6 \cos{\left(3 x \right)}}{\sqrt{1 - x^{2}} \sin{\left(3 x \right)}}\right) \sin^{\operatorname{acos}{\left(x \right)}}{\left(3 x \right)}$$
The third derivative
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/ 3 / 2 \ 2 2 3 \
acos(x) |/ log(sin(3*x)) 3*acos(x)*cos(3*x)\ 27 log(sin(3*x)) / log(sin(3*x)) 3*acos(x)*cos(3*x)\ | x*log(sin(3*x)) 6*cos(3*x) 9*cos (3*x)*acos(x)| 3*x *log(sin(3*x)) 27*cos (3*x) 54*cos (3*x)*acos(x) 54*acos(x)*cos(3*x) 9*x*cos(3*x) |
sin (3*x)*||- ------------- + ------------------| + ----------- - ------------- - 3*|- ------------- + ------------------|*|9*acos(x) + --------------- + -------------------- + -------------------| - ------------------ + --------------------- + -------------------- + ------------------- - --------------------|
|| ________ sin(3*x) | ________ 3/2 | ________ sin(3*x) | | 3/2 ________ 2 | 5/2 ________ 3 sin(3*x) 3/2 |
|| / 2 | / 2 / 2\ | / 2 | | / 2\ / 2 sin (3*x) | / 2\ / 2 2 sin (3*x) / 2\ |
\\ \/ 1 - x / \/ 1 - x \1 - x / \ \/ 1 - x / \ \1 - x / \/ 1 - x *sin(3*x) / \1 - x / \/ 1 - x *sin (3*x) \1 - x / *sin(3*x)/
$$\left(- \frac{3 x^{2} \log{\left(\sin{\left(3 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} - \frac{9 x \cos{\left(3 x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}} \sin{\left(3 x \right)}} + \left(\frac{3 \cos{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{1 - x^{2}}}\right)^{3} - 3 \left(\frac{3 \cos{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{1 - x^{2}}}\right) \left(\frac{x \log{\left(\sin{\left(3 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + 9 \operatorname{acos}{\left(x \right)} + \frac{9 \cos^{2}{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin^{2}{\left(3 x \right)}} + \frac{6 \cos{\left(3 x \right)}}{\sqrt{1 - x^{2}} \sin{\left(3 x \right)}}\right) + \frac{54 \cos{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin{\left(3 x \right)}} + \frac{54 \cos^{3}{\left(3 x \right)} \operatorname{acos}{\left(x \right)}}{\sin^{3}{\left(3 x \right)}} + \frac{27}{\sqrt{1 - x^{2}}} + \frac{27 \cos^{2}{\left(3 x \right)}}{\sqrt{1 - x^{2}} \sin^{2}{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}}\right) \sin^{\operatorname{acos}{\left(x \right)}}{\left(3 x \right)}$$
The graph