91Ó°ÊÓ

The arctan function is the inverse of the tangent function. It is used to find an angle whose tangent is a given number. When you see \(\arctan\left(x\right)\), it means you’re finding the angle \(\theta\) for which \(\tan(\theta) = x\).

In calculus, the arctan function can appear as part of a function whose derivative you need to find. The derivative with respect to \(x\) of \(\arctan(x)\) is \(\frac{1}{1+x^2}\). This derivative comes in handy when differentiating more complex functions involving arctan.

In the presented exercise, the derivative of the function \(f(u, w) = \arctan\left(\frac{u}{w}\right)\) uses the derivative of the arctan function. This step necessitates applying further derivative rules due to the composite nature of the function.

Multivariable calculus extends calculus to functions with more than one variable. Instead of working with a single variable\(x\), like in single-variable calculus, you might deal with variables \(u\) and \(w\) as in this problem.

In problems involving functions of several variables, we often need to find partial derivatives. Partial derivatives express the rate of change of the function with respect to one variable while keeping the others constant.

In \(f(u, w) = \arctan\left(\frac{u}{w}\right)\), finding the partial derivatives with respect to \(u\) and \(w\) means treating one variable as a constant at a time. This process isolates how each variable individually affects the function's value, and it's fundamental in gradient computations and optimizations.

The chain rule is a crucial tool in calculus when dealing with composed functions, such as \(\arctan\left(\frac{u}{w}\right)\). It helps differentiate functions nested inside each other.

When using the chain rule, you find the derivative of the outer function evaluated at the inner function, then multiply it by the derivative of the inner function. For \(\arctan\left(\frac{u}{w}\right)\), the outer function is \(\arctan(x)\) and the inner function is \(x = \frac{u}{w}\). To differentiate with respect to \(u\) or \(w\), apply the chain rule's formula.

• For \(\frac{\partial f}{\partial u}\), treat \(w\) as constant and use \(\frac{d}{du}\left(\frac{u}{w}\right) = \frac{1}{w}\).
• For \(\frac{\partial f}{\partial w}\), treat \(u\) as constant and use \(\frac{d}{dw}\left(\frac{u}{w}\right) = -\frac{u}{w^2}\).

This approach allows for determining how each variable influences the rate of change of the function individually, utilizing their respective roles in the chain rule.