Math Problem Statement

Calculate the derivative using implicit differentiation: ∂w/∂z for the equation x^6w + w^7 + wz^2 + 5yz = 0.

Solution

To solve the problem using implicit differentiation, we are given the equation:

x6w+w7+wz2+5yz=0x^6 w + w^7 + w z^2 + 5yz = 0

We need to find wz\frac{\partial w}{\partial z}. To achieve this, we will differentiate both sides of the equation with respect to zz, treating ww as a function of zz.

Steps for implicit differentiation:

  1. Differentiate x6wx^6w with respect to zz: z(x6w)=x6wz=x6dwdz\frac{\partial}{\partial z} (x^6 w) = x^6 \frac{\partial w}{\partial z} = x^6 \frac{dw}{dz} Since xx is not a function of zz, we treat it as a constant.

  2. Differentiate w7w^7 with respect to zz: z(w7)=7w6wz=7w6dwdz\frac{\partial}{\partial z} (w^7) = 7w^6 \frac{\partial w}{\partial z} = 7w^6 \frac{dw}{dz}

  3. Differentiate wz2wz^2 with respect to zz: Here, we need to use the product rule: z(wz2)=z2dwdz+w2z\frac{\partial}{\partial z} (wz^2) = z^2 \frac{dw}{dz} + w \cdot 2z

  4. Differentiate 5yz5yz with respect to zz: z(5yz)=5y\frac{\partial}{\partial z} (5yz) = 5y Again, yy is not a function of zz, so it is treated as a constant.

Now, putting everything together:

x6dwdz+7w6dwdz+z2dwdz+2wz+5y=0x^6 \frac{dw}{dz} + 7w^6 \frac{dw}{dz} + z^2 \frac{dw}{dz} + 2wz + 5y = 0

Now, we solve for dwdz\frac{dw}{dz}:

dwdz(x6+7w6+z2)=(2wz+5y)\frac{dw}{dz}(x^6 + 7w^6 + z^2) = - (2wz + 5y)

Thus, the derivative is:

dwdz=(2wz+5y)x6+7w6+z2\frac{dw}{dz} = \frac{-(2wz + 5y)}{x^6 + 7w^6 + z^2}

This is the expression for dwdz\frac{dw}{dz}.


Would you like further details or clarifications? Here are five related questions:

  1. How is implicit differentiation different from explicit differentiation?
  2. Can this method be extended to higher-order derivatives?
  3. What is the significance of treating some variables as constants in implicit differentiation?
  4. How does the product rule apply in cases like wz2wz^2?
  5. Can implicit differentiation be used for non-polynomial functions?

Tip: Always remember to apply the chain rule when differentiating implicitly, especially when dealing with products of variables.

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Math Problem Analysis

Mathematical Concepts

Implicit Differentiation
Multivariable Calculus
Product Rule

Formulas

∂/∂z (x^6w) = x^6 ∂w/∂z
∂/∂z (w^7) = 7w^6 ∂w/∂z
∂/∂z (wz^2) = z^2 ∂w/∂z + 2wz
∂/∂z (5yz) = 5y

Theorems

Chain Rule
Product Rule

Suitable Grade Level

Undergraduate Level (Calculus)