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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 -3/x
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Derivative of x^(2*x)
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Derivative of 3*x
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Derivative of 2/sqrt(x)
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Identical expressions
- y=x^sin*x^ three
- y equally x to the power of sinus of multiply by x cubed
- y equally x to the power of sinus of multiply by x to the power of three
- y=xsin*x3
- y=x^sin*x³
- y=x to the power of sin*x to the power of 3
- y=x^sinx^3
- y=xsinx3
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Similar expressions
- x^(sin(x^3))
- y=x^sinx^3
Derivative of y=x^sin*x^3
The solution
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[src]
3 sin (x) x
$$x^{\sin^{3}{\left(x \right)}}$$
/ 3 \ d | sin (x)| --\x / dx
$$\frac{d}{d x} x^{\sin^{3}{\left(x \right)}}$$
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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3 / 3 \
sin (x) |sin (x) 2 |
x *|------- + 3*sin (x)*cos(x)*log(x)|
\ x /
$$x^{\sin^{3}{\left(x \right)}} \left(3 \log{\left(x \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{3}{\left(x \right)}}{x}\right)$$
The second derivative
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3 / 2 2 \
sin (x) |/sin(x) \ 3 sin (x) 2 2 6*cos(x)*sin(x)|
x *||------ + 3*cos(x)*log(x)| *sin (x) - ------- - 3*sin (x)*log(x) + 6*cos (x)*log(x) + ---------------|*sin(x)
|\ x / 2 x |
\ x /
$$x^{\sin^{3}{\left(x \right)}} \left(\left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2} \sin^{3}{\left(x \right)} - 3 \log{\left(x \right)} \sin^{2}{\left(x \right)} + 6 \log{\left(x \right)} \cos^{2}{\left(x \right)} + \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \sin{\left(x \right)}$$
The third derivative
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3 / 3 3 3 2 / 2 \ 2 \
sin (x) |/sin(x) \ 6 9*sin (x) 2*sin (x) 3 2 9*sin (x)*cos(x) 3 /sin(x) \ |sin (x) 2 2 6*cos(x)*sin(x)| 18*cos (x)*sin(x)|
x *||------ + 3*cos(x)*log(x)| *sin (x) - --------- + --------- + 6*cos (x)*log(x) - 21*sin (x)*cos(x)*log(x) - ---------------- - 3*sin (x)*|------ + 3*cos(x)*log(x)|*|------- - 6*cos (x)*log(x) + 3*sin (x)*log(x) - ---------------| + -----------------|
|\ x / x 3 2 \ x / | 2 x | x |
\ x x \ x / /
$$x^{\sin^{3}{\left(x \right)}} \left(\left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{3} \sin^{6}{\left(x \right)} - 3 \cdot \left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \left(3 \log{\left(x \right)} \sin^{2}{\left(x \right)} - 6 \log{\left(x \right)} \cos^{2}{\left(x \right)} - \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \sin^{3}{\left(x \right)} - 21 \log{\left(x \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 6 \log{\left(x \right)} \cos^{3}{\left(x \right)} - \frac{9 \sin^{3}{\left(x \right)}}{x} + \frac{18 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x} - \frac{9 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin^{3}{\left(x \right)}}{x^{3}}\right)$$