Álgebra Ejemplos

$\frac{\log (t)}{\log_{8} (t)}$

Paso 1

Diferencia con la regla del cociente, que establece que $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ es $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ donde $f (t) = \log (t)$ y $g (t) = \log_{8} (t)$ .

$\frac{\log_{8} (t) \frac{d}{d t} [\log (t)] - \log (t) \frac{d}{d t} [\log_{8} (t)]}{{\log_{8}}^{2} (t)}$

Paso 2

La derivada de $\log (t)$ con respecto a $t$ es $\frac{1}{t \ln (10)}$ .

$\frac{\log_{8} (t) \frac{1}{t \ln (10)} - \log (t) \frac{d}{d t} [\log_{8} (t)]}{{\log_{8}}^{2} (t)}$

Paso 3

Combina $\log_{8} (t)$ y $\frac{1}{t \ln (10)}$ .

$\frac{\frac{\log_{8} (t)}{t \ln (10)} - \log (t) \frac{d}{d t} [\log_{8} (t)]}{{\log_{8}}^{2} (t)}$

Paso 4

Multiplica $\frac{\frac{\log_{8} (t)}{t \ln (10)} - \log (t) \frac{d}{d t} [\log_{8} (t)]}{{\log_{8}}^{2} (t)}$ por $\frac{t \ln (10)}{t \ln (10)}$ .

$\frac{t \ln (10)}{t \ln (10)} \cdot \frac{\frac{\log_{8} (t)}{t \ln (10)} - \log (t) \frac{d}{d t} [\log_{8} (t)]}{{\log_{8}}^{2} (t)}$

Paso 5

Simplifica los términos.

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Paso 5.1

Combinar.

$\frac{t \ln (10) (\frac{\log_{8} (t)}{t \ln (10)} - \log (t) \frac{d}{d t} [\log_{8} (t)])}{t \ln (10) {\log_{8}}^{2} (t)}$

Paso 5.2

Aplica la propiedad distributiva.

$\frac{t \ln (10) \frac{\log_{8} (t)}{t \ln (10)} + t \ln (10) (- \log (t) \frac{d}{d t} [\log_{8} (t)])}{t \ln (10) {\log_{8}}^{2} (t)}$

Paso 5.3

Cancela el factor común de $t \ln (10)$ .

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Paso 5.3.1

Cancela el factor común.

$\frac{t \ln (10) \frac{\log_{8} (t)}{t \ln (10)} + t \ln (10) (- \log (t) \frac{d}{d t} [\log_{8} (t)])}{t \ln (10) {\log_{8}}^{2} (t)}$

Paso 5.3.2

Reescribe la expresión.

$\frac{\log_{8} (t) + t \ln (10) (- \log (t) \frac{d}{d t} [\log_{8} (t)])}{t \ln (10) {\log_{8}}^{2} (t)}$

Paso 6

La derivada de $\log_{8} (t)$ con respecto a $t$ es $\frac{1}{t \ln (8)}$ .

$\frac{\log_{8} (t) + t \ln (10) (- \log (t) \frac{1}{t \ln (8)})}{t \ln (10) {\log_{8}}^{2} (t)}$

Paso 7

Simplifica los términos.

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Paso 7.1

Combina $\frac{1}{t \ln (8)}$ y $\log (t)$ .

$\frac{\log_{8} (t) + t \ln (10) (- \frac{\log (t)}{t \ln (8)})}{t \ln (10) {\log_{8}}^{2} (t)}$

Paso 7.2

Combina $t$ y $\frac{\log (t)}{t \ln (8)}$ .

$\frac{\log_{8} (t) + \ln (10) (- \frac{t \log (t)}{t \ln (8)})}{t \ln (10) {\log_{8}}^{2} (t)}$

Paso 7.3

Combina $\ln (10)$ y $\frac{t \log (t)}{t \ln (8)}$ .

$\frac{\log_{8} (t) - \frac{\ln (10) (t \log (t))}{t \ln (8)}}{t \ln (10) {\log_{8}}^{2} (t)}$

Paso 7.4

Cancela el factor común de $t$ .

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Paso 7.4.1

Cancela el factor común.

$\frac{\log_{8} (t) - \frac{\ln (10) t \log (t)}{t \ln (8)}}{t \ln (10) {\log_{8}}^{2} (t)}$

Paso 7.4.2

Reescribe la expresión.

$\frac{\log_{8} (t) - \frac{\ln (10) \log (t)}{\ln (8)}}{t \ln (10) {\log_{8}}^{2} (t)}$

Paso 7.5

Reordena los términos.

$\frac{\log_{8} (t) - \frac{\ln (10) \log (t)}{\ln (8)}}{t {\log_{8}}^{2} (t) \ln (10)}$