Chain Rule with Partial Derivative?

In summary, the problem involves finding the derivatives of z=f(xy) in a neighborhood of the point (1,1,1) with f(x,y) differentiable, given the surface equation x^7y^2+y^4z^6+z^8x^8+9xyz=12. The derivatives are df/dx (1,1) = -24/23 and d^2f/dx^2 (1,1) = unknown. The partial derivative is used by treating y and z as constants. The chain rule is only used if y and/or z are known as functions of x.
  • #1
mshiddensecret
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Homework Statement


Given that the surface x^7y^2+y^4z^6+z^8x^8+9xyz=12 has the equation z=f(xy) in a neighbourhod of the point (1,1,1) with f(x,y) differentiable, find the derivatives.

df/dx (1,1) = ?

d^2f/dx^2 (1,1) = ?

Homework Equations

The Attempt at a Solution


df/dx (1,1) I got -24/23 or
7x^6y^2+6y^4z^5(dz/dx)+8z^7x^8(dz/dx)+8z^8x^7+9yz+9xy(dz/dx)

d^2f/dx^2: I tried doing the same thing and squaring the derivative when they came up again but it didn't work.
 
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  • #2
When you take the partial derivative of f(x,y,z) w.r.t. x, you treat any y's or z's as constants.

For example, if f(x,y,z) = 3x2y + y2z, then

∂ f / ∂ x = 6xy

The chain rule would be used only if you know y and/or z as functions of x.
 

FAQ: Chain Rule with Partial Derivative?

1. What is the chain rule in calculus?

The chain rule is a rule in calculus that allows us to find the derivative of a composite function. A composite function is a function that is made up of two or more functions. The chain rule states that to find the derivative of a composite function, we need to multiply the derivative of the outer function by the derivative of the inner function.

2. How does the chain rule work with partial derivatives?

The chain rule works with partial derivatives in the same way as it does for regular derivatives. The only difference is that when taking partial derivatives, we only consider the variables that we are differentiating with respect to. We treat all other variables as constants and take their derivatives as 0.

3. Why is the chain rule important in multivariable calculus?

The chain rule is important in multivariable calculus because it allows us to find the rate of change of a function with respect to multiple variables. This is useful in many real-life applications such as physics, economics, and engineering, where there are often multiple variables affecting a system.

4. Can the chain rule be applied to higher-order derivatives?

Yes, the chain rule can be applied to higher-order derivatives. The general formula for the chain rule in higher dimensions is known as the multivariable chain rule and involves taking partial derivatives of the outer and inner functions multiple times.

5. How can I practice and improve my understanding of the chain rule with partial derivatives?

The best way to practice and improve your understanding of the chain rule with partial derivatives is to solve various problems and exercises that involve its application. You can also watch online tutorials and videos, work through textbooks and practice problems, or seek help from a tutor or teacher if needed.

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