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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How to use it?
- Derivative of:
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Derivative of -6/x
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Derivative of -2*y
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Derivative of ln(x)+x
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Derivative of cosx-3x
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Identical expressions
- y=(lnx+ one)/(x^2sinx)
- y equally (lnx plus 1) divide by (x squared sinus of x)
- y equally (lnx plus one) divide by (x squared sinus of x)
- y=(lnx+1)/(x2sinx)
- y=lnx+1/x2sinx
- y=(lnx+1)/(x²sinx)
- y=(lnx+1)/(x to the power of 2sinx)
- y=lnx+1/x^2sinx
- y=(lnx+1) divide by (x^2sinx)
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Similar expressions
- y=(lnx-1)/(x^2sinx)
Derivative of y=(lnx+1)/(x^2sinx)
The solution
You have entered
[src]
log(x) + 1 ---------- 2 x *sin(x)
$$\frac{\log{\left(x \right)} + 1}{x^{2} \sin{\left(x \right)}}$$
(log(x) + 1)/((x^2*sin(x)))
Detail solution
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Apply the quotient rule, which is:
and .
To find :
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Differentiate term by term:
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The derivative of the constant is zero.
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The derivative of is .
The result is:
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To find :
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Apply the product rule:
; to find :
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Apply the power rule: goes to
; to find :
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The derivative of sine is cosine:
The result is:
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Now plug in to the quotient rule:
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Now simplify:
The answer is:
The first derivative
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/ 1 \
|---------|
| 2 | / 2 \
\x *sin(x)/ \- x *cos(x) - 2*x*sin(x)/*(log(x) + 1)
----------- + ---------------------------------------
x 4 2
x *sin (x)
$$\frac{\frac{1}{x^{2}} \frac{1}{\sin{\left(x \right)}}}{x} + \frac{\left(- x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)}\right) \left(\log{\left(x \right)} + 1\right)}{x^{4} \sin^{2}{\left(x \right)}}$$
The second derivative
[src]
/ 2 \
|/2 cos(x)\ 2*sin(x) - x *sin(x) + 4*x*cos(x) 2*(2*sin(x) + x*cos(x)) (2*sin(x) + x*cos(x))*cos(x)|
(1 + log(x))*||- + ------|*(2*sin(x) + x*cos(x)) - --------------------------------- + ----------------------- + ----------------------------|
1 \\x sin(x)/ x x sin(x) / 2*(2*sin(x) + x*cos(x))
- - + ---------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------
x sin(x) x*sin(x)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3
x *sin(x)
$$\frac{\frac{\left(\log{\left(x \right)} + 1\right) \left(\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right) + \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x} - \frac{- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}}{x}\right)}{\sin{\left(x \right)}} - \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x \sin{\left(x \right)}} - \frac{1}{x}}{x^{3} \sin{\left(x \right)}}$$
The third derivative
[src]
/ /2 cos(x)\ / 2 \ /2 cos(x)\ /2 cos(x)\ \
| / 2 \ 2 / 2 \ |- + ------|*\2*sin(x) - x *sin(x) + 4*x*cos(x)/ 2*|- + ------|*(2*sin(x) + x*cos(x)) 2 |- + ------|*(2*sin(x) + x*cos(x))*cos(x) / 2 \ |
| | 6 2*cos (x) 4*cos(x)| -6*cos(x) + x *cos(x) + 6*x*sin(x) 6*\2*sin(x) - x *sin(x) + 4*x*cos(x)/ 10*(2*sin(x) + x*cos(x)) \x sin(x)/ \x sin(x)/ 3*cos (x)*(2*sin(x) + x*cos(x)) \x sin(x)/ 3*\2*sin(x) - x *sin(x) + 4*x*cos(x)/*cos(x) 8*(2*sin(x) + x*cos(x))*cos(x)| / 2 \
(1 + log(x))*|2*sin(x) + x*cos(x) + (2*sin(x) + x*cos(x))*|1 + -- + --------- + --------| - ---------------------------------- - ------------------------------------- + ------------------------ - ------------------------------------------------ + ------------------------------------ + ------------------------------- + ----------------------------------------- - -------------------------------------------- + ------------------------------| |/2 cos(x)\ 2*sin(x) - x *sin(x) + 4*x*cos(x) 2*(2*sin(x) + x*cos(x)) (2*sin(x) + x*cos(x))*cos(x)|
| | 2 2 x*sin(x)| x 2 2 x x 2 sin(x) x*sin(x) x*sin(x) | 3*||- + ------|*(2*sin(x) + x*cos(x)) - --------------------------------- + ----------------------- + ----------------------------|
2 \ \ x sin (x) / x x sin (x) / \\x sin(x)/ x x sin(x) / 3*(2*sin(x) + x*cos(x))
-- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- + -----------------------
2 sin(x) x*sin(x) 2
x x *sin(x)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3
x *sin(x)
$$\frac{- \frac{\left(\log{\left(x \right)} + 1\right) \left(x \cos{\left(x \right)} + \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{4 \cos{\left(x \right)}}{x \sin{\left(x \right)}} + \frac{6}{x^{2}}\right) + \frac{3 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right)}{x} + \frac{8 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right) \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x} - \frac{3 \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{x^{2} \cos{\left(x \right)} + 6 x \sin{\left(x \right)} - 6 \cos{\left(x \right)}}{x} + \frac{10 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x^{2}} - \frac{6 \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x^{2}}\right)}{\sin{\left(x \right)}} + \frac{3 \left(\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right) + \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x} - \frac{- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}}{x}\right)}{x \sin{\left(x \right)}} + \frac{3 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x^{2} \sin{\left(x \right)}} + \frac{2}{x^{2}}}{x^{3} \sin{\left(x \right)}}$$
The graph