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- Derivative of a implicit defined function
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- Partial derivative of the function
- Curve tracing functions Step by Step
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- Differential equations Step by Step
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How to use it?
- Derivative of:
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Derivative of e^(sin(1-3*x)^(2))
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Derivative of x×ln(x)
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Derivative of x^3-48x+17
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Derivative of -x+1
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Identical expressions
- y=(sin(3x))^sqrt(x)
- y equally ( sinus of (3x)) to the power of square root of (x)
- y=(sin(3x))^√(x)
- y=(sin(3x))sqrt(x)
- y=sin3xsqrtx
- y=sin3x^sqrtx
Derivative of y=(sin(3x))^sqrt(x)
The solution
You have entered
[src]
___
\/ x
(sin(3*x))
$$\sin^{\sqrt{x}}{\left(3 x \right)}$$
/ ___\ d | \/ x | --\(sin(3*x)) / dx
$$\frac{d}{d x} \sin^{\sqrt{x}}{\left(3 x \right)}$$
Detail solution
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Don't know the steps in finding this derivative.
But the derivative is
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Now simplify:
The answer is:
The first derivative
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___ / ___ \
\/ x |log(sin(3*x)) 3*\/ x *cos(3*x)|
(sin(3*x)) *|------------- + ----------------|
| ___ sin(3*x) |
\ 2*\/ x /
$$\left(\frac{3 \sqrt{x} \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{2 \sqrt{x}}\right) \sin^{\sqrt{x}}{\left(3 x \right)}$$
The second derivative
[src]
/ 2 \
| / ___ \ |
| |log(sin(3*x)) 6*\/ x *cos(3*x)| |
| |------------- + ----------------| |
___ | | ___ sin(3*x) | ___ 2 |
\/ x | ___ \ \/ x / log(sin(3*x)) 9*\/ x *cos (3*x) 3*cos(3*x) |
(sin(3*x)) *|- 9*\/ x + ----------------------------------- - ------------- - ----------------- + --------------|
| 4 3/2 2 ___ |
\ 4*x sin (3*x) \/ x *sin(3*x)/
$$\left(- 9 \sqrt{x} - \frac{9 \sqrt{x} \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} + \frac{\left(\frac{6 \sqrt{x} \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{x}}\right)^{2}}{4} + \frac{3 \cos{\left(3 x \right)}}{\sqrt{x} \sin{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{4 x^{\frac{3}{2}}}\right) \sin^{\sqrt{x}}{\left(3 x \right)}$$
The third derivative
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/ 3 \
| / ___ \ / ___ \ / ___ 2 \ |
| |log(sin(3*x)) 6*\/ x *cos(3*x)| |log(sin(3*x)) 6*\/ x *cos(3*x)| | ___ log(sin(3*x)) 12*cos(3*x) 36*\/ x *cos (3*x)| |
| |------------- + ----------------| 3*|------------- + ----------------|*|36*\/ x + ------------- - -------------- + ------------------| |
___ | | ___ sin(3*x) | | ___ sin(3*x) | | 3/2 ___ 2 | ___ 3 ___ 2 |
\/ x | 27 \ \/ x / \ \/ x / \ x \/ x *sin(3*x) sin (3*x) / 3*log(sin(3*x)) 54*\/ x *cos (3*x) 54*\/ x *cos(3*x) 27*cos (3*x) 9*cos(3*x) |
(sin(3*x)) *|- ------- + ----------------------------------- - ----------------------------------------------------------------------------------------------------- + --------------- + ------------------ + ----------------- - ----------------- - ---------------|
| ___ 8 8 5/2 3 sin(3*x) ___ 2 3/2 |
\ 2*\/ x 8*x sin (3*x) 2*\/ x *sin (3*x) 4*x *sin(3*x)/
$$\left(\frac{54 \sqrt{x} \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \frac{54 \sqrt{x} \cos^{3}{\left(3 x \right)}}{\sin^{3}{\left(3 x \right)}} + \frac{\left(\frac{6 \sqrt{x} \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{x}}\right)^{3}}{8} - \frac{3 \cdot \left(\frac{6 \sqrt{x} \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\sqrt{x}}\right) \left(36 \sqrt{x} + \frac{36 \sqrt{x} \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} - \frac{12 \cos{\left(3 x \right)}}{\sqrt{x} \sin{\left(3 x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{x^{\frac{3}{2}}}\right)}{8} - \frac{27}{2 \sqrt{x}} - \frac{27 \cos^{2}{\left(3 x \right)}}{2 \sqrt{x} \sin^{2}{\left(3 x \right)}} - \frac{9 \cos{\left(3 x \right)}}{4 x^{\frac{3}{2}} \sin{\left(3 x \right)}} + \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}}{8 x^{\frac{5}{2}}}\right) \sin^{\sqrt{x}}{\left(3 x \right)}$$
The graph