Explained: "Find both first partial derivatives. z=6x−x2y+8y2" - Plainmath

Josalynn

Answered question

2021-02-05

Find both first partial derivatives.

z = 6 x - x^{2} y + 8 y^{2}

Answer & Explanation

Elberte

Skilled2021-02-06Added 95 answers

Step 1: Given that
Find both first partial derivatives.

z = 6 x - x^{2} y + 8 y^{2}

Step 2: Solve
We have,

z = 6 x - x^{2} y + 8 y^{2}

Differentiating partially both sides w.r.t x we obtain,

\frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (6 x - x^{2} y + 8 y^{2})

= \frac{\partial}{\partial x} (6 x) - \frac{\partial}{\partial x} (x^{2} y) + \frac{\partial}{\partial x} (8 y^{2})

= 6 - 2 x y + 0

= 6 - 2 x y

Similarly, Differentiating partially the given function both sides w.r.t y we obtain,

\frac{\partial z}{\partial y} = \frac{\partial}{\partial y} (6 x - x^{2} y + 8 y^{2})

= \frac{\partial}{\partial y} (6 x) - \frac{\partial}{\partial y} (x^{2} y) + \frac{\partial}{\partial y} (8 y^{2})

= 0 - x^{2} + 16 y

= 16 y - x^{2}

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