Sr Examen
Otras calculadoras
- Derivada de una función implícitamente dada
- Derivada de una función paramétrica
- Derivada parcial de la función
- Análisis de la función gráfica
- Integrales paso a paso
- Ecuaciones diferenciales paso a paso
- Límites paso por paso
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¿Cómo usar?
- Derivada de:
-
Derivada de 6/x
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Derivada de e^(x/2)
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Derivada de (x-1)/(x+1)
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Derivada de -3/x
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Expresiones idénticas
- y=(lnx)/((x²+ cuatro)²sin³x)
- y es igual a (lnx) dividir por ((x² más 4)² seno de ³x)
- y es igual a (lnx) dividir por ((x² más cuatro)² seno de ³x)
- y=lnx/x²+4²sin³x
- y=(lnx) dividir por ((x²+4)²sin³x)
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Expresiones semejantes
- y=(lnx)/((x²-4)²sin³x)
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Expresiones con funciones
- lnx
- lnx-x
- lnx-x^3
- lnx^sinx
- lnx+sinx
- lnx/e^x^2
Derivada de y=(lnx)/((x²+4)²sin³x)
Solución
Ha introducido
[src]
log(x)
-----------------
2
/ 2 \ 3
\x + 4/ *sin (x)
$$\frac{\log{\left(x \right)}}{\left(x^{2} + 4\right)^{2} \sin^{3}{\left(x \right)}}$$
log(x)/(((x^2 + 4)^2*sin(x)^3))
Solución detallada
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Se aplica la regla de la derivada parcial:
y .
Para calcular :
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Derivado es .
Para calcular :
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Se aplica la regla de la derivada de una multiplicación:
; calculamos :
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Sustituimos .
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Según el principio, aplicamos: tenemos
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Luego se aplica una cadena de reglas. Multiplicamos por :
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diferenciamos miembro por miembro:
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La derivada de una constante es igual a cero.
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Según el principio, aplicamos: tenemos
Como resultado de:
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Como resultado de la secuencia de reglas:
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; calculamos :
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Sustituimos .
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Según el principio, aplicamos: tenemos
-
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Luego se aplica una cadena de reglas. Multiplicamos por :
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La derivada del seno es igual al coseno:
Como resultado de la secuencia de reglas:
-
Como resultado de:
Ahora aplicamos la regla de la derivada de una divesión:
Simplificamos:
Respuesta:
Primera derivada
[src]
/ 1 \
|-----------------|
| 2 | / 2 \
|/ 2 \ 3 | | 3 / 2 \ / 2 \ 2 |
\\x + 4/ *sin (x)/ \- 4*x*sin (x)*\x + 4/ - 3*\x + 4/ *sin (x)*cos(x)/*log(x)
------------------- + ------------------------------------------------------------
x 4
/ 2 \ 6
\x + 4/ *sin (x)
$$\frac{\left(- 4 x \left(x^{2} + 4\right) \sin^{3}{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \right)}}{\left(x^{2} + 4\right)^{4} \sin^{6}{\left(x \right)}} + \frac{\frac{1}{\left(x^{2} + 4\right)^{2}} \frac{1}{\sin^{3}{\left(x \right)}}}{x}$$
Segunda derivada
[src]
/ 2 2 \
| / 2\ 2 2 / 2\ / 2\ 2 2 2 / 2\ / / 2\ \ / / 2\ \|
|/3*cos(x) 4*x \ / / 2\ \ - 3*\4 + x / *sin (x) + 4*sin (x)*\4 + x / + 6*\4 + x / *cos (x) + 8*x *sin (x) + 24*x*\4 + x /*cos(x)*sin(x) 3*\3*\4 + x /*cos(x) + 4*x*sin(x)/*cos(x) 4*x*\3*\4 + x /*cos(x) + 4*x*sin(x)/|
||-------- + ------|*\3*\4 + x /*cos(x) + 4*x*sin(x)/ - ------------------------------------------------------------------------------------------------------------- + ----------------------------------------- + ------------------------------------|*log(x)
|| sin(x) 2| / 2\ sin(x) 2 | / / 2\ \
1 \\ 4 + x / \4 + x /*sin(x) 4 + x / 2*\3*\4 + x /*cos(x) + 4*x*sin(x)/
- -- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------
2 / 2\ / 2\
x \4 + x /*sin(x) x*\4 + x /*sin(x)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
/ 2\ 3
\4 + x / *sin (x)
$$\frac{\frac{\left(\frac{4 x \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x^{2} + 4} + \left(\frac{4 x}{x^{2} + 4} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) + \frac{3 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{8 x^{2} \sin^{2}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} + 6 \left(x^{2} + 4\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}}\right) \log{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}} - \frac{2 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x \left(x^{2} + 4\right) \sin{\left(x \right)}} - \frac{1}{x^{2}}}{\left(x^{2} + 4\right)^{2} \sin^{3}{\left(x \right)}}$$
Tercera derivada
[src]
/ / 2 2 \ \
| /3*cos(x) 4*x \ | / 2\ 2 2 / 2\ / 2\ 2 2 2 / 2\ | /3*cos(x) 4*x \ / / 2\ \ /3*cos(x) 4*x \ / / 2\ \ |
| / 2 2 \ |-------- + ------|*\- 3*\4 + x / *sin (x) + 4*sin (x)*\4 + x / + 6*\4 + x / *cos (x) + 8*x *sin (x) + 24*x*\4 + x /*cos(x)*sin(x)/ / 2 2 \ / 2 2 \ 3*|-------- + ------|*\3*\4 + x /*cos(x) + 4*x*sin(x)/*cos(x) 4*x*|-------- + ------|*\3*\4 + x /*cos(x) + 4*x*sin(x)/ |
| / 2 2 \ / / 2\ \ | / 2\ 3 3 3 / 2\ / 2\ 2 2 / 2\ 2 2 2 / 2\ | 2 / / 2\ \ 2 / / 2\ \ | sin(x) 2| | / 2\ 2 2 / 2\ / 2\ 2 2 2 / 2\ | | / 2\ 2 2 / 2\ / 2\ 2 2 2 / 2\ | | sin(x) 2| | sin(x) 2| / / 2\ \ | / 2 2 \
|/ / 2\ \ | 4 12*cos (x) 24*x 24*x*cos(x) | 4*\3*\4 + x /*cos(x) + 4*x*sin(x)/ / 2\ 3*\2*\4 + x / *cos (x) + 8*x*sin (x) - 12*x*sin (x)*\4 + x / - 7*\4 + x / *sin (x)*cos(x) + 12*sin (x)*\4 + x /*cos(x) + 24*x *sin (x)*cos(x) + 24*x*cos (x)*\4 + x /*sin(x)/ 21*cos (x)*\3*\4 + x /*cos(x) + 4*x*sin(x)/ 40*x *\3*\4 + x /*cos(x) + 4*x*sin(x)/ \ 4 + x / 12*x*\- 3*\4 + x / *sin (x) + 4*sin (x)*\4 + x / + 6*\4 + x / *cos (x) + 8*x *sin (x) + 24*x*\4 + x /*cos(x)*sin(x)/ 9*\- 3*\4 + x / *sin (x) + 4*sin (x)*\4 + x / + 6*\4 + x / *cos (x) + 8*x *sin (x) + 24*x*\4 + x /*cos(x)*sin(x)/*cos(x) \ 4 + x / \ 4 + x / 48*x*\3*\4 + x /*cos(x) + 4*x*sin(x)/*cos(x)| | / 2\ 2 2 / 2\ / 2\ 2 2 2 / 2\ / / 2\ \ / / 2\ \|
|\3*\4 + x /*cos(x) + 4*x*sin(x)/*|3 - ------ + ---------- + --------- + ---------------| - ---------------------------------- + 9*\4 + x /*cos(x) + 12*x*sin(x) + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------- + -------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------- + -------------------------------------------------------- + --------------------------------------------|*log(x) |/3*cos(x) 4*x \ / / 2\ \ - 3*\4 + x / *sin (x) + 4*sin (x)*\4 + x / + 6*\4 + x / *cos (x) + 8*x *sin (x) + 24*x*\4 + x /*cos(x)*sin(x) 3*\3*\4 + x /*cos(x) + 4*x*sin(x)/*cos(x) 4*x*\3*\4 + x /*cos(x) + 4*x*sin(x)/|
| | 2 2 2 / 2\ | 2 / 2\ 2 2 2 / 2\ 2 / 2\ 2 sin(x) 2 / 2\ | 3*||-------- + ------|*\3*\4 + x /*cos(x) + 4*x*sin(x)/ - ------------------------------------------------------------------------------------------------------------- + ----------------------------------------- + ------------------------------------|
| | 4 + x sin (x) / 2\ \4 + x /*sin(x)| 4 + x \4 + x /*sin (x) sin (x) / 2\ \4 + x /*sin(x) / 2\ \4 + x /*sin (x) 4 + x \4 + x /*sin(x) | || sin(x) 2| / 2\ sin(x) 2 | / / 2\ \
2 \ \ \4 + x / / \4 + x / \4 + x / *sin(x) / \\ 4 + x / \4 + x /*sin(x) 4 + x / 3*\3*\4 + x /*cos(x) + 4*x*sin(x)/
-- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------
3 / 2\ / 2\ 2 / 2\
x \4 + x /*sin(x) x*\4 + x /*sin(x) x *\4 + x /*sin(x)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
/ 2\ 3
\4 + x / *sin (x)
$$\frac{- \frac{\left(\frac{40 x^{2} \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{\left(x^{2} + 4\right)^{2}} + 12 x \sin{\left(x \right)} + \frac{4 x \left(\frac{4 x}{x^{2} + 4} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x^{2} + 4} + \frac{48 x \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}} - \frac{12 x \left(8 x^{2} \sin^{2}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} + 6 \left(x^{2} + 4\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)}\right)}{\left(x^{2} + 4\right)^{2} \sin{\left(x \right)}} + 9 \left(x^{2} + 4\right) \cos{\left(x \right)} + \frac{3 \left(\frac{4 x}{x^{2} + 4} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \left(\frac{24 x^{2}}{\left(x^{2} + 4\right)^{2}} + \frac{24 x \cos{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}} + 3 + \frac{12 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{4}{x^{2} + 4}\right) + \frac{21 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{\left(\frac{4 x}{x^{2} + 4} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(8 x^{2} \sin^{2}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} + 6 \left(x^{2} + 4\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)}\right)}{\left(x^{2} + 4\right) \sin{\left(x \right)}} - \frac{4 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x^{2} + 4} - \frac{9 \left(8 x^{2} \sin^{2}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} + 6 \left(x^{2} + 4\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{\left(x^{2} + 4\right) \sin^{2}{\left(x \right)}} + \frac{3 \left(24 x^{2} \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 12 x \left(x^{2} + 4\right) \sin^{3}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)} + 8 x \sin^{3}{\left(x \right)} - 7 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 2 \left(x^{2} + 4\right)^{2} \cos^{3}{\left(x \right)} + 12 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)}{\left(x^{2} + 4\right) \sin^{2}{\left(x \right)}}\right) \log{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}} + \frac{3 \left(\frac{4 x \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x^{2} + 4} + \left(\frac{4 x}{x^{2} + 4} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) + \frac{3 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{8 x^{2} \sin^{2}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} + 6 \left(x^{2} + 4\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}}\right)}{x \left(x^{2} + 4\right) \sin{\left(x \right)}} + \frac{3 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x^{2} \left(x^{2} + 4\right) \sin{\left(x \right)}} + \frac{2}{x^{3}}}{\left(x^{2} + 4\right)^{2} \sin^{3}{\left(x \right)}}$$
Gráfico