Find 𝑓𝑥 and 𝑓𝑦 for 𝑓(𝑥,𝑦)=(1+𝑥−𝑦)𝑒𝑥+2𝑦 Question 9Select one: a. 𝑓𝑥=(−1−2𝑥+𝑦)𝑒𝑥+2𝑦,𝑓𝑦=(3−4𝑥+2𝑦)𝑒𝑥+2𝑦. b. 𝑓𝑥=(2+𝑥−2𝑦)𝑒𝑥+2𝑦,𝑓𝑦=(2𝑥−4𝑦)𝑒𝑥+2𝑦. c. None of these d. 𝑓𝑥=(2+𝑥−𝑦)𝑒𝑥+2𝑦,𝑓𝑦=(1+2𝑥−2𝑦)𝑒𝑥+2𝑦. e. 𝑓𝑥=(−𝑥+𝑦)𝑒𝑥+2𝑦,𝑓𝑦=(3−2𝑥+2𝑦)𝑒𝑥+2𝑦.

To find the partial derivatives of the function 𝑓(𝑥,𝑦)=(1+𝑥−𝑦)𝑒𝑥+2𝑦, we need to differentiate the function with respect to x and y separately.

Step 1: Find 𝑓𝑥 The derivative of the function with respect to x is found by differentiating each term of the function with respect to x.

The derivative of (1+𝑥−𝑦)𝑒𝑥 with respect to x is (1+𝑥−𝑦)𝑒𝑥 + (𝑒𝑥)(1) = (1+𝑥−𝑦+1)𝑒𝑥 = (2+𝑥−𝑦)𝑒𝑥.

The derivative of 2𝑦 with respect to x is 0, because y is treated as a constant when differentiating with respect to x.

So, 𝑓𝑥 = (2+𝑥−𝑦)𝑒𝑥.

Step 2: Find 𝑓𝑦 The derivative of the function with respect to y is found by differentiating each term of the function with respect to y.

The derivative of (1+𝑥−𝑦)𝑒𝑥 with respect to y is (𝑒𝑥)(-1) = -𝑒𝑥.

The derivative of 2𝑦 with respect to y is 2.

So, 𝑓𝑦 = -𝑒𝑥 + 2.

None of the options given match these results, so the answer is (c) None of these.