What is the implicit derivative of #13=2xe^y-x^3e^(2y) #?

Question

Emily Ammerman · Accepted Answer

#dy/dx=(3x^2e^y-2)/(2x(1-x^2e^y))#

This will require Implicit Differentiation, Chain Rule, and Product Rule. The most important thing to remember is that since we're differentiating with respect to #x#, anything with #y# will spit out a #dy/dx# term.

#d/dx[13=2xe^y-x^3e^(2y)]#

Let's first find the derivative of each inner part.

#d/dx[2xe^y]=2d/dx[xe^y]=2(e^y+xe^ydy/dx)=2e^y+2xe^ydy/dx#

#d/dx[x^3e^(2y)]=3x^2e^(2y)+2x^3e^(2y)dy/dx#

Back to the original:

#0=2e^y+2xe^ydy/dx-3x^2e^(2y)-2x^3e^(2y)dy/dx#

#3x^2e^(2y)-2e^y=dy/dx(2xe^y-2x^3e^(2y))#

#dy/dx=(3x^2e^(2y)-2e^y)/(2xe^y-2x^3e^(2y))#

#dy/dx=(e^y(3x^2e^y-2))/(e^y(2x-2x^3e^y))#

#dy/dx=(3x^2e^y-2)/(2x(1-x^2e^y))#

Christopher Allers · Answer

undefined To find the implicit derivative of the given equation $ 13 = 2xe^y - x^3e^{2y} $, we differentiate both sides of the equation with respect to $ x $ using the chain rule and product rule.

Differentiating $ 13 $ with respect to $ x $ gives $ 0 $, and for the right-hand side, we have:
$$ \frac{d}{dx} (2xe^y) = 2e^y + 2xe^y \frac{dy}{dx} $$
$$ \frac{d}{dx} (x^3e^{2y}) = 3x^2e^{2y} + x^3(2e^{2y}) \frac{dy}{dx} $$

Putting it all together and solving for $ \frac{dy}{dx} $:
$$ 0 = 2e^y + 2xe^y \frac{dy}{dx} - (3x^2e^{2y} + 2x^3e^{2y} \frac{dy}{dx}) $$
$$ \frac{dy}{dx} (2xe^y - 2x^3e^{2y}) = 3x^2e^{2y} - 2e^y $$
$$ \frac{dy}{dx} = \frac{3x^2e^{2y} - 2e^y}{2xe^y - 2x^3e^{2y}} $$