Step by step solution to "Find the first partial derivatives of the following..." - Plainmath

Mylo O'Moore

Answered question

2021-03-09

Find the first partial derivatives of the following functions.

f (x, y) = x e^{y}

Answer & Explanation

jlo2niT

Skilled2021-03-10Added 96 answers

Step 1
To find the first-order partial derivatives of the function,

f (x, y) = x e^{y}

Step 2
The partial derivatives with respect to x is given by,

f_{x} (x, y) = lim_{h \to 0} \frac{f (x + h, y) - f (x, y)}{h} = lim_{h \to 0} \frac{(x + h) e^{y} - x e^{y}}{h} = lim_{h \to 0} \frac{x e^{y} + h e^{y} - x e^{y}}{h} = lim_{h \to 0} \frac{h e^{y}}{h}

= lim_{h \to 0} e^{y} = e^{y}

The partial derivatives with respect to y is given by,

f_{y} (x, y) = lim_{k \to 0} \frac{f (x, y + k) - f (x, y)}{h} = lim_{k \to 0} \frac{x e^{y + k} - x e^{y}}{k} = lim_{k \to 0} \frac{x e^{y} (e^{k} - 1)}{k} = x e^{y} lim_{k \to 0} \frac{e^{k} - 1}{k}

= x e^{y} lim_{k \to 0} e^{k} = x e^{y} (1) = x e^{y}

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