$\text { Find } \frac { d y } { d x } \text {. }$
$x = 7 e ^ { \theta }$
$y = e ^ { - \frac { \theta } { 7 } }$
A)$\frac { d y } { d x } = \frac { 1 } { 49 } e ^ { - \frac { d } { 7 } }$
B) $\frac { d y } { d x } = 49 e ^ { - 7 \theta }$
C)$\frac { d y } { d x } = - \frac { 1 } { 7 } e ^ { - \frac { 8 } { 7 } \theta }$
D) $\frac { d y } { d x } = 7 e ^ { - 7 \theta }$
E)$\frac { d y } { d x } = - \frac { 1 } { 49 } e ^ { - \frac { 8 } { 7 } \theta }$

Question

$\text { Find } \frac { d y } { d x } \text {. }$ 
$x = 7 e ^ { \theta }$
$y = e ^ { - \frac { \theta } { 7 } }$
A)$\frac { d y } { d x } = \frac { 1 } { 49 } e ^ { - \frac { d } { 7 } }$
B) $\frac { d y } { d x } = 49 e ^ { - 7 \theta }$
C)$\frac { d y } { d x } = - \frac { 1 } { 7 } e ^ { - \frac { 8 } { 7 } \theta }$
D) $\frac { d y } { d x } = 7 e ^ { - 7 \theta }$
E)$\frac { d y } { d x } = - \frac { 1 } { 49 } e ^ { - \frac { 8 } { 7 } \theta }$

Quizplus · Accepted Answer

To find $\frac{dy}{dx}$, we use the chain rule: $\frac{dy}{dx} = \frac{dy}{d\theta} \cdot \frac{d\theta}{dx}$. First, differentiate $y$ with respect to $\theta$: $\frac{dy}{d\theta} = -\frac{1}{7}e^{-\frac{\theta}{7}}$. Then, differentiate $x$ with respect to $\theta$ and invert it to find $\frac{d\theta}{dx}$: $x = 7e^{\theta}$ implies $\frac{dx}{d\theta} = 7e^{\theta}$, so $\frac{d\theta}{dx} = \frac{1}{7e^{\theta}}$. Multiplying these gives $\frac{dy}{dx} = -\frac{1}{7}e^{-\frac{\theta}{7}} \cdot \frac{1}{7e^{\theta}} = -\frac{1}{49}e^{-\frac{8\theta}{7}}$.

Find dydx. \text { Find } \frac { d y } { d x } \text {. } Find dxdy​.

$\text { Find } \frac { d y } { d x } \text {. }$