Let f(x, y) = (-(5x + y))^8. Then find \frac{\partial^2 f}{\partial x \partial y}

Question:

Let {eq}f(x, y) = (-(5x + y))^8{/eq}. Then find {eq}\frac{\partial^2 f}{\partial x \partial y}{/eq}

Partial Derivatives:

If f(x, y) is a function in two variables, then

(i) {eq}\dfrac{\partial f}{\partial x} {/eq} is the partial derivative of f with respect to x. i.e, we have to differentiate f with respect to x, keeping y as constant.

(ii) {eq}\dfrac{\partial f}{\partial y} {/eq} is the partial derivative of f with respect to y. i.e., we have to differentiate f with respect to y, keeping x as constant.

We can calculate the second-order partial derivatives using the following formulas:

{eq}\begin{aligned} f_{x x} &=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right)=\frac{\partial^{2} f}{\partial x^{2}} \\ f_{y y} &=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right)=\frac{\partial^{2} f}{\partial y^{2}} \\ f_{y x} &=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)=\frac{\partial^{2} f}{\partial x \partial y} \\ f_{x y} &=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)=\frac{\partial^{2} f}{\partial y \partial x} \end{aligned} {/eq}

Answer and Explanation: 1

The given function is,

{eq}f(x, y) = (-(5x + y))^8= (-1)^8 (5x+y)^8 = (5x+y)^8 {/eq}

{eq}\dfrac{\partial f}{\partial y} = \dfrac{\partial }{\partial y} ((5x+y)^8) = 8(5x+y)^7 \dfrac{\partial }{\partial y} (5x+y) = 8(5x+y)^7 (1) = 8(5x+y)^7 {/eq}

(We used chain rule here).

Now we find the required partial derivative.

{eq}\begin{align} \dfrac{\partial ^2f}{\partial x \partial y} &= \dfrac{\partial }{\partial x} \left( \dfrac{\partial f}{\partial y} \right) \\ & = \dfrac{\partial }{\partial x}(8(5x+y)^7) \\ & = 8 (7 (5x+y)^6) \dfrac{\partial }{\partial x} (5x+y) & \text{(By chain rule)} \\ & = 56(5x+y)^6 (5) \\ & = 280(5x+y)^6 \end{align} {/eq}

Therefore, {eq}\mathbf{\dfrac{\partial ^2f}{\partial x \partial y}=280(5x+y)^6} {/eq}


Learn more about this topic:

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Higher-Order Partial Derivatives Definition & Examples

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Chapter 14 / Lesson 2
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In this lesson, we define the partial derivative and then extend this concept to find higher-order partial derivatives. Examples are used to expand your knowledge and skill.


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