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A partial derivative is a derivative where we hold all but one variable constant. It measures how a function changes as one particular variable changes while others remain fixed. This concept is vital in multivariable calculus where functions of more than one variable are studied. When taking a partial derivative of a function like \(f(x, y)\), you treat the other variables as constants while differentiating with respect to one of them.

For example, if you have a function like \(f(x, y) = e^{\sin y}\), the partial derivative with respect to \(x\) is zero. This is because \(f\) does not depend on \(x\), therefore \(\frac{\partial f}{\partial x} = 0\). This highlights an important aspect of partial derivatives: they focus on changes in one direction, ignoring others.

Similarly, for \(y\), you calculate \(\frac{\partial f}{\partial y}\) by differentiating as if \(x\) were a constant, which in this case results in applying the chain rule, as the next section explains.

The chain rule in calculus is a formula for computing the derivative of a composite function. It's particularly useful when dealing with functions of multiple variables that are defined as the exponentials, trigonometric, or other composite forms.

When you have a function like \(f(x, y) = e^{\sin y}\), and you need to find the partial derivative with respect to \(y\), you encounter a composite function situation. Here, \(u = \sin y\) and \(f(u) = e^u\). The chain rule tells us to first take the derivative of the outer function \(e^u\) with respect to \(u\), which is \(e^u\), and then multiply it by the derivative of the inner function \(\sin y\) with respect to \(y\), which is \(\cos y\).

• The derivative of \(e^{u}\) with respect to \(u\) is \(e^{u}\).
• The derivative of \(u = \sin y\) with respect to \(y\) is \(\cos y\).

Combining these using the chain rule gives \(\frac{\partial f}{\partial y} = e^{\sin y} \cdot \cos y\). This process allows us to systematically and accurately differentiate composite functions.

Evaluation of functions refers to the process of determining the value of a function for specific inputs. This is often a final step after deriving expressions through differentiation or solving equations, where you substitute specific values for the variables to calculate the result.

In our example with the function \(f(x, y) = e^{\sin y}\) and the point \((0, \pi)\), we evaluated the partial derivatives. Since \(\frac{\partial f}{\partial x}\) was zero, it did not depend on the values of \(x\) or \(y\). However, for \(\frac{\partial f}{\partial y}\), you need to substitute \(y = \pi\).

This substitution turns the expression \(e^{\sin y} \cdot \cos y\) into \(e^{\sin \pi} \cdot \cos \pi\), simplifying to \(-1\):

• \(\sin \pi = 0\), so \(e^{0} = 1\).
• \(\cos \pi = -1\).

Thus, multiplying these results gives \(-1\). Evaluating at such points simplifies complex problems by providing concrete numbers to substitute back, helping to understand the behavior of the function at specific locations.