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Partial differentiation is a technique used to find the derivative of a function with multiple variables by treating other variables as constants. It is especially useful in physics and engineering, where functions often depend on more than one variable.
For the given function, we have two variables: 't' and 'x'. To find how the function changes with time, we differentiate only with respect to 't', keeping 'x' constant. Similarly, to see how it changes with position 'x', we differentiate with respect to 'x', treating 't' as a constant.
This method allows us to analyze changes in different dimensions separately, providing a deeper understanding of the dynamics at play.

When functions are composed of other functions, the chain rule is an essential mathematical tool for differentiation. It allows us to find the derivative of composite functions step-by-step.
In our function, consider how \(z = e^{-t} \sin \frac{x}{c}\) is composed. For differentiation with respect to 't', \(e^{-t}\) is the outer function, and \(\sin \frac{x}{c}\) is constant with respect to 't'. To differentiate, the derivative of \(e^{-t}\) is found, resulting in \(-e^{-t}\sin \frac{x}{c}\).
When differentiating with respect to 'x', the \(\sin \frac{x}{c}\) function must react to changes in 'x'. Here, use the chain rule again by differentiating \(\sin(y)\) as \(\cos(y) \,dy/dx\) where \(y = \frac{x}{c}\), later resulting in \(\frac{\partial^{2} z}{\partial x^{2}} = -e^{-t} \sin \frac{x}{c} / c^{2}\).Understanding the chain rule facilitates breaking down complex multi-variable functions into a manageable process.

Once derivatives are calculated, it's essential to verify if a proposed function satisfies a particular equation, like the heat equation in this case. Verification is basically substituting these derivatives into the equation to see if both sides match.
After finding \(\frac{\partial z}{\partial t}\) and \(\frac{\partial^{2} z}{\partial x^{2}}\), substitute them back into the heat equation \(\frac{\partial z}{\partial t} = c^{2}\frac{\partial^{2} z}{\partial x^{2}}\). If both sides are equal, the original function is confirmed as a solution.
This step ensures that all differentiation was performed correctly and confirms the function as a valid solution to the problem posed by the equation.