Find the derivative of y with respect to the independent variable.
-$$y = 3 ^{\ln 4 t}$$
A) $3 ^{\ln 4 t}$
B) $\frac { 4 \ln 3 } { t }$
C) $\frac { \ln 3 } { t } 3 \ln 4 t$
D) $\frac { 4 \ln 3 } { t } 3 \ln 4 t$

Question

Quizplus · Accepted Answer

Using the chain rule, we have:

$$y' = \frac{d}{dt}\left(3^{\ln(4t)}\right) = 3^{\ln(4t)}\cdot \frac {d}{dt}\ln(4t)$$

Since $\frac{d}{dx}\ln(x) = \frac{1}{x}$, we have:

$$y' = 3^{\ln(4t)}\cdot \frac {1}{4t}\cdot 4 = \frac{3^{\ln(4t)}\ln(4t)}{t}$$

Now we use the property that $a^{\log_a(b)}=b$ to simplify $3^{\ln(4t)}$:

$$y' = \frac{(4t)(\ln(3)\ln(4t))}{t} = \ln(3)\ln(4t) = \frac{\ln(3)}{t}3\ln(4t)$$

Thus, the answer is option C.

Find the Derivative of Y with Respect to the Independent y=3ln⁡4ty = 3 ^{\ln 4 t}y=3ln4t

Find the Derivative of Y with Respect to the Independent $y = 3 ^{\ln 4 t}$