Let $e ^ { x } = 3 \sin y$ . Use implicit differentiation to find $d y / d x$ when $( x , y ) = ( 1,0 )$ .
A) $- \frac { 1 } { 3 } e$

B) $\frac { 1 } { 3 } e$
C) $- \frac { 1 } { 2 } e$
D) $\frac { 1 } { 2 } e$
E) $- e$
F) $e$

G) $- 2 e$

H) $2e$

Question

Let $e ^ { x } = 3 \sin y$ . Use implicit differentiation to find $d y / d x$ when $( x , y ) = ( 1,0 )$ .
A) $- \frac { 1 } { 3 } e$

B) $\frac { 1 } { 3 } e$ 
C) $- \frac { 1 } { 2 } e$ 
D) $\frac { 1 } { 2 } e$ 
E) $- e$ 
F) $e$

G) $- 2 e$

H) $2e$

Quizplus · Accepted Answer

Using implicit differentiation, we have:$$\frac{d}{dx}e^x = \frac{d}{dx}3\sin y$$$$e^x = 3\cos y \frac{dy}{dx}$$$$\frac{dy}{dx} = \frac{e^x}{3\cos y}$$At $(1,0)$, we have:$$e^1 = 3\cos 0$$$$e = 3$$So,$$\frac{dy}{dx} = \frac{e}{3\cos y} = \frac{e}{3} = \boxed{\frac{1}{3}e}$$

Let ex=3sin⁡ye ^ { x } = 3 \sin yex=3siny Use Implicit Differentiation to Find

Let $e ^ { x } = 3 \sin y$ Use Implicit Differentiation to Find