91Ó°ÊÓ

The chain rule is a fundamental concept in calculus used for computing the derivative of a composite function. If you have two functions, say, \( g(u) \) and \( u=f(x) \), the chain rule helps in differentiating the composite function \( g(f(x)) \). To apply the chain rule, you follow these steps:

• Differentiating the outer function \( g(u) \) with respect to the intermediate variable \( u \).
• Then, differentiating the inner function \( f(x) \) with respect to the original variable \( x \).
• Multiplying these derivatives together gives the derivative of the composite function: \( \frac{d}{dx}[g(f(x))]= g' (f(x)) * f' (x) \) .

In our exercise, we used the chain rule to differentiate the term \( \sqrt{x^{2}+y^{2}} \) with respect to \( x \). Here, the inner function is \( u = x^{2} + y^{2} \) , and the outer function is \( \sqrt{u} \) . Following the application of the chain rule, we first differentiate the outer function and later the inner function.

Ordinary differentiation involves finding the rate at which a function changes with respect to one of its variables. It primarily deals with functions of one variable. For instance, if you have a function \( f(x) \), its ordinary derivative \( \frac{df}{dx} \) indicates how \( f \) changes as \( x \) changes. In our exercise, ordinary differentiation is used when finding the partial derivative of a multi-variable function, but treating other variables as constants. For example, to find \( \frac{\partial}{\partial x}f(x, y) \), we differentiate with respect to \( x \) and treat \( y \) as a constant, effectively transforming the function temporarily into a single-variable function.

A function of multiple variables takes more than one input. For example, \( f(x, y) \) takes two inputs, \( x \) and \( y \). The output value of \( f(x, y) \) depends on both these inputs. These types of functions e increasingly common in fields such as physics and economics. Key aspects to consider with multivariable functions include:

• Each variable can independently influence the function's output.
• You can fix one variable and investigate the function's behavior with respect to the other variable.
• Common visualization tools include 3D plots, where the third dimension represents the function's output.

In our exercise, the function \( f(x, y) = 4y^{3} + \sqrt{x^{2} + y^{2}} \) is a function of two variables. To find partial derivatives, we observe how the function changes with respect to one variable, keeping the other constant.

Partial differentiation involves finding the derivative of a function of multiple variables with respect to one of these variables. The steps to do this include:

• Identify the variable with respect to which you are differentiating.
• Treat all other variables as constants.
• Apply the differentiation rules (product rule, chain rule, etc.) as if it were a single-variable function.
• Simplify the resulting expression.

Let's revisiting our exercise:

• We started by identifying the partial derivative \( \frac{\partial f}{\partial x} \) .
• Recognized the terms that depend on \( x \) -- namely, \( \sqrt{x^{2} + y^{2}} \).
• Other terms like \( 4y^{3} \) were treated as constants.
• Next, with the chain rule, differentiated \( \sqrt{x^{2} + y^{2}} \) with respect to \( x \).
• Finally, combined and simplified the results, yielding \( \frac{x}{\sqrt{x^{2} + y^{2}}} \).

By following these steps, you accurately find how a function's output changes with one variable while holding others constant.