Partial fraction derivative

Question

Need help on this problem appreciated it thank you very much.

It is talking about the partial fraction derivative of this function below.

Please provide steps so I can understand how it needs to be done.

find ∂z/∂y given the following:

−

Answer 1

Maybe you referred to partial derivative, since partial fractions is a method to integrate a function.

In any case, you start by massaging your equation in order to have the variable z on the left hand side of an equality and all the other terms on the right hand side.

In this case, you should get z= 9/(xy) + (1/2)*x⁴y⁷ - e^6xy/(2xy).

We now can evaluate the partial derivative of z with respect to y which will be the sum of the derivatives of the above three terms.

First term: 9/(xy)

The derivative of this term with respect to y is given by -9/(xy²)

Second term: (1/2)*x⁴y⁷

The derivative of this term with respect to y is given by (7/2)x⁴y⁶

Third term: e^xy/(2xy)

The derivative of this term with respect to y needs to be evaluated with the Quotient Rule and is given by the ration where at the numerator you have 6xye^6xy-e^6xyand at the denominator you have 2xy²

Putting all these things together you get the final result

-9/(xy²)+e(7/2)x⁴y⁶ -(1/2xy²)*(6xye^6xy-e^6xy)

Best,

Davide

Answer 2

18=2xyz−x⁵y⁸+e^(6xy)

For partial derivatives note that any variables other than y and z act as constants.

Taking partial derivatives with respect to y (recall all composite functions require the chain rule)

0 = 2xz +2xy(∂z/∂y) -8x⁵y⁷ + 6xe^6xy

-2xy(∂z/∂y) = 2xz -8x⁵y⁷ + 6xe^6xy

∂z/∂y =- (2xz -8x⁵y⁷ + 6xe^6xy)/2xy