How do you implicitly differentiate #y= xy^2 + x ^2 e^(x y) #?

Question

How do you implicitly differentiate #y= xy^2 + x  ^2 e^(x y)   #?

Ezra Adams · Accepted Answer

#frac{"d"y}{"d"x} = frac{y^2 + x^2ye^(xy) + 2xe^(xy)}{1 - x^3e^(xy) - 2xy}#

This problem requires knowledge of the Product Rule and the Chain Rule.

#y = xy^2 + x^2e^(xy)#

Differentiate both sides w.r.t. #x#.

#frac{"d"}{"d"x}(y) = frac{"d"}{"d"x}(xy^2 + x^2e^(xy))#

#frac{"d"y}{"d"x} = frac{"d"}{"d"x}(xy^2) + frac{"d"}{"d"x}(x^2e^(xy))#

#frac{"d"y}{"d"x} = (xfrac{"d"}{"d"x}(y^2) + y^2frac{"d"}{"d"x}(x)) #

#+ (x^2frac{"d"}{"d"x}(e^(xy)) + e^(xy)frac{"d"}{"d"x}(x^2))#

#frac{"d"y}{"d"x} = (xfrac{"d"}{"d"y}(y^2)frac{"d"y}{"d"x} + y^2) #

#+ (x^2e^(xy)frac{"d"}{"d"x}(xy) + e^(xy)(2x))#

#frac{"d"y}{"d"x} = (x(2y)frac{"d"y}{"d"x} + y^2) #

#+ (x^2e^(xy)(yfrac{"d"}{"d"x}(x) + xfrac{"d"}{"d"x}(y)) + 2xe^(xy))#

#frac{"d"y}{"d"x} = (2xyfrac{"d"y}{"d"x} + y^2) #

#+ (x^2e^(xy)(y + xfrac{"d"y}{"d"x}) + 2xe^(xy))#

#frac{"d"y}{"d"x} = 2xyfrac{"d"y}{"d"x} + y^2 + x^2ye^(xy) + x^3e^(xy)frac{"d"y}{"d"x} + 2xe^(xy)#

Now make #frac{"d"y}{"d"x}# the subject of formula.

#frac{"d"y}{"d"x} - x^3e^(xy)frac{"d"y}{"d"x} - 2xyfrac{"d"y}{"d"x} = y^2 + x^2ye^(xy) + 2xe^(xy)#

#frac{"d"y}{"d"x}(1 - x^3e^(xy) - 2xy) = y^2 + x^2ye^(xy) + 2xe^(xy)#

#frac{"d"y}{"d"x} = frac{y^2 + x^2ye^(xy) + 2xe^(xy)}{1 - x^3e^(xy) - 2xy}#

Christopher Allers · Answer

undefined To implicitly differentiate $ y = xy^2 + x^2 e^{xy} $, follow these steps:

1. Differentiate both sides of the equation with respect to $ x $.
2. Apply the product rule when differentiating terms involving $ x $ and $ y $.
3. Use the chain rule when differentiating terms involving $ e^{xy} $.
4. Solve for $ \frac{dy}{dx} $ to find the derivative.

The implicit differentiation of $ y = xy^2 + x^2 e^{xy} $ yields:

$$ \frac{dy}{dx} = y^2 + 2xy + 2xe^{xy} + x^2 y e^{xy} $$