Multivariable chain rule question

In summary: C}{\partial u^2}\dfrac{du}{dx}\right)u'+\left(\frac{\partial^2C}{\partial u\partial v}\dfrac{du}{dx}\right)v'+\left(\frac{\partial^2C}{\partial v\partial u}\dfrac{dv}{dx}\right)u'+\left(\frac{\partial^2C}{\partial v^2}\dfrac{dv}{dx}\right)v'In summary, to find the second derivative of the function C(u,v) evaluated at u=F(x) and v=G(x), we use the product and chain rules to write out the expressions for \d
  • #1
willy0625
6
0
I am trying to find the second derivative of the function

[tex] C:[0,1]^{2} \rightarrow [0,1] ,\quad \mbox{defined by }C=C(u,v) [/tex]

evaluated at

[tex]u=F(x)=1-\exp(-\lambda_{1} x),\quad \lambda_{1} \geq 0 [/tex]

and

[tex]v=G(x)=1-\exp(-\lambda_{2} x),\quad \lambda_{2} \geq 0[/tex]

First I work out the first derivative which is

[tex]\dfrac{dC}{dx} = \dfrac{\partial C}{\partial u}\dfrac{du}{dx}+\dfrac{\partial C}{\partial v}\dfrac{dv}{dx}[/tex]

Now, I have trouble working out the second derivative because it looks like I have to used the chain rule again and there is product rule which involves differentiating

[tex]\dfrac{du}{dx}[/tex]

with respect to u(and v)??I would appreciate any reply. Thank you guys.
 
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  • #2
Please put tex-tags around your latex expressions.

[tex]\dfrac{dC}{dx} = \dfrac{\partial C}{\partial u}u'+\dfrac{\partial C}{\partial v}v'[/tex]

First use the product (and sum) rule:

[tex]\dfrac{d^2C}{dx^2} = \left(\frac{d}{dx}\dfrac{\partial C}{\partial u}\right)u'+\frac{\partial C}{\partial u}u''+\left(\frac{d}{dx}\dfrac{\partial C}{\partial v}\right)v'+\frac{\partial C}{\partial v}v''[/tex]

Now use the chain rule to write out the epressions

[tex]\left(\frac{d}{dx}\dfrac{\partial C}{\partial u}\right)[/tex]
[tex]\left(\frac{d}{dx}\dfrac{\partial C}{\partial v}\right)[/tex]
 

1. What is the multivariable chain rule?

The multivariable chain rule is a mathematical concept that allows us to find the derivative of a composite function with multiple variables. It is an extension of the chain rule, which is used to find the derivative of a single variable function.

2. When should the multivariable chain rule be used?

The multivariable chain rule should be used when dealing with a composite function that has multiple variables. It is particularly useful in multivariable calculus and in solving optimization problems.

3. How do you apply the multivariable chain rule?

To apply the multivariable chain rule, you first need to identify the inner and outer functions of the composite function. Then, you can take the derivative of the outer function while treating the inner function as a constant. Finally, you multiply the derivative of the outer function with the derivative of the inner function.

4. Can the multivariable chain rule be applied to any composite function?

Yes, the multivariable chain rule can be applied to any composite function, as long as the inner and outer functions are differentiable. However, it may become increasingly complex and difficult to apply as the number of variables and functions in the composite function increases.

5. What are some common mistakes when using the multivariable chain rule?

Some common mistakes when using the multivariable chain rule include forgetting to apply the chain rule and its derivatives correctly, mixing up the inner and outer functions, and not simplifying the final expression. It is important to carefully follow the steps and practice regularly to avoid these mistakes.

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