2) (g(u)=3^{4u - 1}): find (g(t))
2) (g(u)=3^{4u - 1}): find (g(t))
Answer
Explanation:
Step1: Recall the derivative formula
The derivative of $a^{u}$ with respect to $u$ is $a^{u}\ln a$. Here $a = 3$ and $u=4t - 1$.
Step2: Apply the chain - rule
If $y = 3^{4t - 1}$, let $u = 4t-1$. Then $\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}$. First, $\frac{dy}{du}=3^{u}\ln 3$ and $\frac{du}{dt}=4$. Substituting $u = 4t - 1$ back in, we get $g^{\prime}(t)=3^{4t - 1}\ln 3\times4$.
Answer:
$g^{\prime}(t)=4\cdot3^{4t - 1}\ln 3$