Mister Exam
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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 x^12
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Derivative of x^2*e^(3-2*x)*log(2)^x
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Derivative of -(x^2+324)/x
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Derivative of x^3-1/5*x^2+2*x-4
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Identical expressions
- y=sqrt(ln*sin(3x))
- y equally square root of (ln multiply by sinus of (3x))
- y=√(ln*sin(3x))
- y=sqrt(lnsin(3x))
- y=sqrtlnsin3x
Derivative of y=sqrt(ln*sin(3x))
The solution
You have entered
[src]
_________________ \/ log(x)*sin(3*x)
$$\sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}}$$
d / _________________\ --\\/ log(x)*sin(3*x) / dx
$$\frac{d}{d x} \sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}}$$
Detail solution
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Let .
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Apply the power rule: goes to
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Then, apply the chain rule. Multiply by :
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Apply the product rule:
; to find :
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The derivative of is .
; to find :
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Let .
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The derivative of sine is cosine:
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Then, apply the chain rule. Multiply by :
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The derivative of a constant times a function is the constant times the derivative of the function.
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Apply the power rule: goes to
So, the result is:
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The result of the chain rule is:
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The result is:
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The result of the chain rule is:
Now simplify:
The answer is:
The first derivative
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_________________ /sin(3*x) 3*cos(3*x)*log(x)\
\/ log(x)*sin(3*x) *|-------- + -----------------|
\ 2*x 2 /
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log(x)*sin(3*x)
$$\frac{\sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}} \left(\frac{3 \log{\left(x \right)} \cos{\left(3 x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{2 x}\right)}{\log{\left(x \right)} \sin{\left(3 x \right)}}$$
The second derivative
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/ 2\
| /sin(3*x) \ sin(3*x) /sin(3*x) \ |
| 3*|-------- + 3*cos(3*x)*log(x)|*cos(3*x) -------- + 3*cos(3*x)*log(x) |-------- + 3*cos(3*x)*log(x)| |
_________________ |3*cos(3*x) 9*log(x)*sin(3*x) sin(3*x) \ x / x \ x / |
\/ log(x)*sin(3*x) *|---------- - ----------------- - -------- - ----------------------------------------- - ---------------------------- + -------------------------------|
| x 2 2 2*sin(3*x) 2*x*log(x) 4*log(x)*sin(3*x) |
\ 2*x /
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log(x)*sin(3*x)
$$\frac{\sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}} \left(- \frac{9 \log{\left(x \right)} \sin{\left(3 x \right)}}{2} - \frac{3 \cdot \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right) \cos{\left(3 x \right)}}{2 \sin{\left(3 x \right)}} + \frac{\left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right)^{2}}{4 \log{\left(x \right)} \sin{\left(3 x \right)}} + \frac{3 \cos{\left(3 x \right)}}{x} - \frac{3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}}{2 x \log{\left(x \right)}} - \frac{\sin{\left(3 x \right)}}{2 x^{2}}\right)}{\log{\left(x \right)} \sin{\left(3 x \right)}}$$
The third derivative
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/ sin(3*x) 6*cos(3*x) /sin(3*x) 6*cos(3*x) \ 3 2 2 /sin(3*x) \ /sin(3*x) 6*cos(3*x) \ \
| -------- - ---------- + 9*log(x)*sin(3*x) sin(3*x) sin(3*x) 3*|-------- - ---------- + 9*log(x)*sin(3*x)|*cos(3*x) 2 /sin(3*x) \ /sin(3*x) \ /sin(3*x) \ /sin(3*x) \ 3*|-------- + 3*cos(3*x)*log(x)|*|-------- - ---------- + 9*log(x)*sin(3*x)| /sin(3*x) \ |
| 2 x -------- + 3*cos(3*x)*log(x) -------- + 3*cos(3*x)*log(x) | 2 x | 9*cos (3*x)*|-------- + 3*cos(3*x)*log(x)| |-------- + 3*cos(3*x)*log(x)| 9*|-------- + 3*cos(3*x)*log(x)| *cos(3*x) 3*|-------- + 3*cos(3*x)*log(x)| \ x / | 2 x | 3*|-------- + 3*cos(3*x)*log(x)|*cos(3*x)|
_________________ |sin(3*x) 9*sin(3*x) 9*cos(3*x) x x x \ x / \ x / \ x / \ x / \ x / \ x / \ x / |
\/ log(x)*sin(3*x) *|-------- - ---------- - ---------- + ----------------------------------------- + ---------------------------- + ---------------------------- + ------------------------------------------------------ + ------------------------------------------ + ------------------------------- - ------------------------------------------ - --------------------------------- - ---------------------------------------------------------------------------- + -----------------------------------------|
| 3 x 2 x*log(x) 2 2 2 sin(3*x) 2 2 2 2 2 4*log(x)*sin(3*x) x*log(x)*sin(3*x) |
\ x 2*x x *log (x) 2*x *log(x) sin (3*x) 8*log (x)*sin (3*x) 4*log(x)*sin (3*x) 4*x*log (x)*sin(3*x) /
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log(x)*sin(3*x)
$$\frac{\sqrt{\log{\left(x \right)} \sin{\left(3 x \right)}} \left(\frac{9 \cdot \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right) \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} + \frac{3 \cdot \left(9 \log{\left(x \right)} \sin{\left(3 x \right)} - \frac{6 \cos{\left(3 x \right)}}{x} + \frac{\sin{\left(3 x \right)}}{x^{2}}\right) \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} - \frac{9 \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right)^{2} \cos{\left(3 x \right)}}{4 \log{\left(x \right)} \sin^{2}{\left(3 x \right)}} - \frac{3 \cdot \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right) \left(9 \log{\left(x \right)} \sin{\left(3 x \right)} - \frac{6 \cos{\left(3 x \right)}}{x} + \frac{\sin{\left(3 x \right)}}{x^{2}}\right)}{4 \log{\left(x \right)} \sin{\left(3 x \right)}} - \frac{9 \sin{\left(3 x \right)}}{x} + \frac{\left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right)^{3}}{8 \log{\left(x \right)}^{2} \sin^{2}{\left(3 x \right)}} + \frac{3 \cdot \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right) \cos{\left(3 x \right)}}{x \log{\left(x \right)} \sin{\left(3 x \right)}} + \frac{9 \log{\left(x \right)} \sin{\left(3 x \right)} - \frac{6 \cos{\left(3 x \right)}}{x} + \frac{\sin{\left(3 x \right)}}{x^{2}}}{x \log{\left(x \right)}} - \frac{9 \cos{\left(3 x \right)}}{2 x^{2}} - \frac{3 \left(3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}\right)^{2}}{4 x \log{\left(x \right)}^{2} \sin{\left(3 x \right)}} + \frac{3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}}{2 x^{2} \log{\left(x \right)}} + \frac{\sin{\left(3 x \right)}}{x^{3}} + \frac{3 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{\sin{\left(3 x \right)}}{x}}{x^{2} \log{\left(x \right)}^{2}}\right)}{\log{\left(x \right)} \sin{\left(3 x \right)}}$$
The graph