use implicit differentiation to find $\frac{dy}{dx}$. $cos(4y^{2}) - 8x = 9e^{y}$ $\frac{dy}{dx}=square$…

use implicit differentiation to find $\frac{dy}{dx}$. $cos(4y^{2}) - 8x = 9e^{y}$ $\frac{dy}{dx}=square$ (use parentheses to clearly denote the argument of each function.)

use implicit differentiation to find $\frac{dy}{dx}$. $cos(4y^{2}) - 8x = 9e^{y}$ $\frac{dy}{dx}=square$ (use parentheses to clearly denote the argument of each function.)

Answer

Answer:

$\frac{8}{- 8y\sin(4y^{2})-9e^{y}}$

Explanation:

Step1: Differentiate both sides

Differentiate $\cos(4y^{2})-8x = 9e^{y}$ with respect to $x$. The derivative of $\cos(4y^{2})$ using the chain - rule is $-\sin(4y^{2})\cdot8y\frac{dy}{dx}$, the derivative of $-8x$ is $-8$, and the derivative of $9e^{y}$ is $9e^{y}\frac{dy}{dx}$. So we have $-\sin(4y^{2})\cdot8y\frac{dy}{dx}-8 = 9e^{y}\frac{dy}{dx}$.

Step2: Isolate $\frac{dy}{dx}$

Move all terms with $\frac{dy}{dx}$ to one side: $-8y\sin(4y^{2})\frac{dy}{dx}-9e^{y}\frac{dy}{dx}=8$. Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(-8y\sin(4y^{2}) - 9e^{y})=8$.

Step3: Solve for $\frac{dy}{dx}$

Divide both sides by $-8y\sin(4y^{2})-9e^{y}$ to get $\frac{dy}{dx}=\frac{8}{- 8y\sin(4y^{2})-9e^{y}}$.