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The chain rule is a fundamental tool in calculus used to differentiate composite functions. When dealing with multiple variables, this rule helps us find how changes in one variable affect a function, considering that other variables might also be changing.

To use the chain rule, imagine breaking a complex function into simpler parts, then finding derivatives of each part. For a function like \(f(x, y, z) = \tanh(x + 2y + 3z)\), which involves a composition of the tangent hyperbolic function and a linear combination of variables, the chain rule simplifies the process of differentiation.

The basic idea is:

• Differentiate the outer function: \(\tanh(u)\) gives \(\text{sech}^2(u)\).
• Then, multiply it by the derivative of the inner function with respect to the variable of interest.

For instance, to find \(f_x\), differentiate \(u = x + 2y + 3z\) with respect to \(x\), which equals 1, resulting in \(f_x = \text{sech}^2(u) \cdot 1\). For \(f_y\) and \(f_z\), follow similarly by differentiating \(u\) with respect to \(y\) and \(z\), leading to multipliers of 2 and 3, respectively.

Hyperbolic functions, like \(\tanh, \sinh,\) and \(\cosh\), are analogs of trigonometric functions but for a hyperbola instead of a circle. The \(\tanh\) function is particularly interesting in calculus due to its unique properties and symmetry.

The \(\tanh(u)\) function, or hyperbolic tangent, is defined as:

• \(\tanh(u) = \frac{\sinh(u)}{\cosh(u)}\)
• Where \(\sinh(u) = \frac{e^u - e^{-u}}{2}\) and \(\cosh(u) = \frac{e^u + e^{-u}}{2}\)

This gives \(\tanh(u)\) similar properties to \(\sin(u)\), but while the sine function oscillates between -1 and 1, \(\tanh(u)\) approaches these limits asymptotically.

When taking derivatives, the nice property is that \(\frac{d}{du}\tanh(u) = \text{sech}^2(u)\), which means differentiating \(\tanh(u)\) results in another hyperbolic function, the hyperbolic secant squared, making these operations very smooth.

Multivariable calculus expands the principles of calculus to functions involving more than one variable. It allows us to study how functions change as each input variable changes, while holding others constant.

In the problem \(f(x, y, z) = \tanh(x + 2y + 3z)\), the task is to find partial derivatives \(f_x, f_y,\) and \(f_z\). Partial derivatives show the rate of change of a function with respect to one variable, ignoring the effect of changes in other variables.

• For \(f_x\), treat \(y\) and \(z\) as constants and differentiate with respect to \(x\).
• Similarly, for \(f_y\) and \(f_z\), treat the other variables as constants.

This method of holding variables static while differentiating one at a time simplifies complex relations and aids in constructing tools like gradients and tangent planes. Multivariable functions often require partial derivatives for modeling real-world phenomena such as thermodynamics, economics, and more, where multiple factors influence outcomes simultaneously.