$$\begin{array}{rl}f(x,y)& ={e}^{xy}\\ g(x,y)& =f(\mathrm{sin}({x}^{2}+y),{x}^{3}+2y+1)\end{array}$$

Let’s compute $\frac{{\mathrm{\partial}}^{2}g}{\mathrm{\partial}{x}^{2}}(0,0)$

2k1ablakrh0
2022-09-24
Answered

How to apply the chain rule to a double partial derivative of a multivariable function?

$$\begin{array}{rl}f(x,y)& ={e}^{xy}\\ g(x,y)& =f(\mathrm{sin}({x}^{2}+y),{x}^{3}+2y+1)\end{array}$$

Let’s compute $\frac{{\mathrm{\partial}}^{2}g}{\mathrm{\partial}{x}^{2}}(0,0)$

$$\begin{array}{rl}f(x,y)& ={e}^{xy}\\ g(x,y)& =f(\mathrm{sin}({x}^{2}+y),{x}^{3}+2y+1)\end{array}$$

Let’s compute $\frac{{\mathrm{\partial}}^{2}g}{\mathrm{\partial}{x}^{2}}(0,0)$

You can still ask an expert for help

xjiaminhoxy4

Answered 2022-09-25
Author has **9** answers

If we let $f$ be defined as $f(u,v)={e}^{uv}$ instead, for clarity, then

$$\frac{\mathrm{\partial}g}{\mathrm{\partial}x}=\frac{\mathrm{\partial}f}{\mathrm{\partial}u}\frac{\mathrm{\partial}u}{\mathrm{\partial}x}+\frac{\mathrm{\partial}f}{\mathrm{\partial}v}\frac{\mathrm{\partial}v}{\mathrm{\partial}x}$$

Once you've calculated the first partial derivative, you repeat the above on said partial derivative to get the second derivative.

$$\frac{\mathrm{\partial}g}{\mathrm{\partial}x}=\frac{\mathrm{\partial}f}{\mathrm{\partial}u}\frac{\mathrm{\partial}u}{\mathrm{\partial}x}+\frac{\mathrm{\partial}f}{\mathrm{\partial}v}\frac{\mathrm{\partial}v}{\mathrm{\partial}x}$$

Once you've calculated the first partial derivative, you repeat the above on said partial derivative to get the second derivative.

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Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years. In about how many years will human teeth be 90% of their present size?

Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years. In about how many years will human teeth be 90% of their present size?

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see the equation as attached here:-

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see the equation as attached here:-

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determine if the following functions are differentiable at $(x,y)=(0,0)$.

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$$g(x,y)={e}^{{|x|}^{3}y}$$

$$f(x,y)=\sqrt{|xy|}$$

$$g(x,y)={e}^{{|x|}^{3}y}$$