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The chain rule is a crucial tool in calculus for finding derivatives of composite functions. A composite function is one where you have a function inside another function, like a nested doll.

For example, with our function \( z = \cos(x^2 + y^2) \), you have \( z = \cos(u) \) and \( u = x^2 + y^2 \). The chain rule helps us differentiate this by breaking down the steps:

1. First, differentiate the outer function \( \cos(u) \) with respect to \( u \), which gives \( -\sin(u) \).
2. Then, multiply by the derivative of the inner function \( u \) with respect to the variable of interest.

This process ensures you account for all changes within the composite function, making the chain rule a powerful tool whenever dealing with functions inside functions.

A partial derivative with respect to \( x \) means that you're finding the derivative while keeping other variables constant, like \( y \).

This approach helps isolate the effect of changing one variable at a time.

For \( z = \cos(x^2 + y^2) \), applying the partial derivative:

• Keep \( y \) constant.
• Use the chain rule to differentiate: \( z_x = -\sin(x^2 + y^2) \cdot \frac{\partial}{\partial x}(x^2 + y^2) \).
• This results in \( -2x \sin(x^2 + y^2) \).

This calculation shows how changes in \( x \) affect \( z \) when \( y \) is fixed.

Similarly, the partial derivative with respect to \( y \) involves changing \( y \) while keeping \( x \) constant.

This reveals how the function \( z \) responds to changes in \( y \) alone.

For the same function, \( z = \cos(x^2 + y^2) \), we calculate:

• Hold \( x \) constant.
• Apply the chain rule: \( z_y = -\sin(x^2 + y^2) \cdot \frac{\partial}{\partial y}(x^2 + y^2) \).
• This leads to \( -2y \sin(x^2 + y^2) \).

These steps highlight the impact of changes in \( y \) on the overall function \( z \), offering insights into how the function shifts with varying inputs.