Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.
-$$y = e ^ { 3 x / 2 }$$
A) $y = e ^ { u } ; u = 3 x / 2 ; \frac { d y } { d x } = \frac { 2 } { 3 } e ^ { 3 x / 2 }$
B) $y = e ^ { u } ; u = 3 x / 2 ; \frac { d y } { d x } = \frac { 2 } { 3 } x e ^ { 3 x / 2 }$
C) $y = e ^ { u } ; u = 3 x / 2 ; \frac { d y } { d x } = \frac { 3 } { 2 } e ^ { 3 x / 2 }$
D) $y = e ^ { u } ; u = 3 x / 2 ; \frac { d y } { d x } = \frac { 3 } { 2 } x e ^ { 3 x / 2 }$

Question

Quizplus · Accepted Answer

First, we rewrite $y = e^{3x/2}$ as $y = e^u$ where $u = 3x/2$. Then, using the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Since $\frac{dy}{du} = e^u$ and $\frac{du}{dx} = \frac{3}{2}$, we find $\frac{dy}{dx} = e^u \cdot \frac{3}{2} = \frac{3}{2} e^{3x/2}$.

Write the Function in the Form Y = F(u) and U