Implicit Differentiation

[Icebreaker]

$$f(x,y,z,u,v)=xe^y+uz-\cos v=2$$
$$g(x,y,z,u,v)=u\cos y+x^2v-yz^2=1$$

I need to find $$u_z$$. When I try to do it by implicitly differentiating and solving the equation, I get 2 contradictory answers. If I try the formula, i.e.

$$f_z + f_uu_z + f_vv_z = 0$$
$$g_z + g_uu_z + g_vv_z = 0$$

I get an answer, but I'm not sure if it's right, since it does not equal to the answer I get when I differentiate implicitly. Any help?

Also I'm not entirely sure if my "formula" is right. Maybe this formula is just implicit differentiation, I haven't looked into it.

 

[quantumdude]

2 things:

1. Please show how you started, and where you got stuck.

2. Please stop posting homework problems in the Math section. They should go in the Homework section.

Thanks.

 

[quicknote]

Implicit differentiation is a powerful tool in multivariable calculus that allows us to find the derivative of a function that is not explicitly defined in terms of a single variable. In this case, we have two functions, f(x,y,z,u,v) and g(x,y,z,u,v), each with five variables. When we use implicit differentiation to find the derivative of one of these functions with respect to a particular variable, we must treat all other variables as functions of the variable we are differentiating with respect to.

In this case, we are trying to find the derivative of f with respect to z, or f_z. This means that we must treat x, y, u, and v as functions of z. Similarly, when finding g_z, we must treat x, y, u, and v as functions of z.

The formula you are using is correct, as it is the general formula for implicit differentiation. However, there may be some errors in your calculations that are causing the contradictory answers. It is important to double check your work and make sure you are using the correct chain rule when differentiating each term in the function.

I would recommend reviewing the steps for implicit differentiation and carefully checking your calculations to ensure accuracy. If you are still getting contradictory answers, it may be helpful to seek assistance from a tutor or professor to identify any potential errors.