How do you find #(delf(x,y))/(dely)# and #(delf(x,y))/(delx)# of #f(x,y)=(yx^22y)/(2ye^x+y^3)#, using the quotient rule?
To partially differentiate, we treat one variable as a constant.
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To find ( \frac{\partial f(x,y)}{\partial y} ) and ( \frac{\partial f(x,y)}{\partial x} ) of ( f(x,y) = \frac{yx^2  2y}{2ye^x + y^{3}} ) using the quotient rule, follow these steps:

For ( \frac{\partial f(x,y)}{\partial y} ): Apply the quotient rule, which states that for functions ( u(x) ) and ( v(x) ), the derivative of ( \frac{u(x)}{v(x)} ) is given by ( \frac{u'v  uv'}{v^2} ). Let ( u(y) = yx^2  2y ) and ( v(y) = 2ye^x + y^{3} ). Find ( \frac{\partial u}{\partial y} ) and ( \frac{\partial v}{\partial y} ). Then apply the quotient rule formula.

For ( \frac{\partial f(x,y)}{\partial x} ): Follow similar steps but this time differentiate with respect to ( x ).
Here are the steps in detail:

For ( \frac{\partial f(x,y)}{\partial y} ): ( u(y) = yx^2  2y ) ( v(y) = 2ye^x + y^{3} ) ( \frac{\partial u}{\partial y} = x^2  2 ) ( \frac{\partial v}{\partial y} = 2e^x  (3)y^{4} ) Apply the quotient rule: ( \frac{\partial f}{\partial y} = \frac{(x^2  2)(2ye^x + y^{3})  (yx^2  2y)(2e^x  (3)y^{4})}{(2ye^x + y^{3})^2} )

For ( \frac{\partial f(x,y)}{\partial x} ): ( u(y) = yx^2  2y ) ( v(y) = 2ye^x + y^{3} ) ( \frac{\partial u}{\partial x} = 2yx ) ( \frac{\partial v}{\partial x} = 2ye^x ) Apply the quotient rule: ( \frac{\partial f}{\partial x} = \frac{(2yx)(2ye^x + y^{3})  (yx^2  2y)(2ye^x)}{(2ye^x + y^{3})^2} )
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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