Problem 2 - Calculus of a Single Variable

Question

Find \(d y / d x\) by implicit differentiation. \(x^{2}-y^{2}=25\)

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = \frac{x}{y}\)

Step 1: Differentiate both sides with respect to x

Differentiate \(x^{2}\) with respect to x and \(-y^{2}\) with respect to x. The result is \(2x - 2y \frac{dy}{dx} = 0\). The chain rule is applied in differentiating \(-y^{2}\), which is why there is \(\frac{dy}{dx}\) in the result.

Step 2: Express \(\frac{dy}{dx}\) in terms of x and y

Rearrange the equation resulting from step 1 to make \(\frac{dy}{dx}\) the subject of the formula. The answer is \(\frac{dy}{dx} = \frac{x}{y}\).

Key Concepts

Chain Rule

When we encounter functions within functions during differentiation, a technique called the chain rule comes into play. In calculus, it's one of the most pivotal rules, allowing us to differentiate composite functions. Imagine you have two functions, one nested within the other. Instead of trying to simplify the nested functions down to a single formula, which can be complex or sometimes impossible, the chain rule helps us split the differentiation process into manageable steps.

Here's a simplified way to understand it: if we have a function h(x) that is composed of two functions, say f(g(x)), then the derivative h'(x) is found by multiplying the derivative of the outer function f at g(x) by the derivative of the inner function g at x. Symbolically, it's expressed as \(h'(x) = f'(g(x)) \cdot g'(x)\). In the context of implicit differentiation, as shown in the original exercise, we use the chain rule to differentiate terms like \(y^2\) with respect to x, hence the appearance of \(\frac{dy}{dx}\) in the differentiated equation.

Differentiation

Differentiation is the process of finding the derivative of a function. It tells us how the function's value changes as its input changes, in essence providing the rate of change or the slope of the function at any given point. The derivative represents the instantaneous rate of change, akin to finding how quickly a car is moving at a precise moment.

Applying Differentiation to Polynomials

When we differentiate basic polynomials, we apply simple rules like the power rule, which states that for any term \(x^n\), its derivative is \(nx^{n-1}\). This rule is clearly seen in our exercise when the term \(x^2\) is differentiated to become \(2x\).

Calculus

Calculus is a branch of mathematics that studies how things change. It's built on two foundational concepts: differentiation and integration. While differentiation focuses on the rate at which quantities change, integration is about accumulation of quantities. Calculus can describe everything from simple motion to the changes in the stock market, and it's essential for understanding the physical world around us.

Calculus in Real-World Applications

Calculus helps us calculate the trajectory of a planet, the growth of populations, or the curve of a wave. Implicit differentiation, part of the calculus toolbox, allows us to handle equations where the variables can't be easily separated, offering a method to find derivatives when functions are intertwined.

Derivative

A derivative represents an instantaneous rate of change – for example, how fast a car is accelerating at a particular moment. It is the fundamental concept derived from differentiation. In our exercise, taking the derivative of both sides of the equation with respect to x involves finding \(dx/dx\), which simplifies to 1, and \(d(-y^2)/dx\), which applies the chain rule as it’s seen as a composition of \(y\) with itself squared.

Understanding Derivatives of Implicit Functions

For implicit functions—equations where y is not isolated on one side—finding the derivative requires implicit differentiation. This is crucial because not all relationships between variables are straightforward, and sometimes separating the variables is impractical or impossible. Implicit differentiation respects the interdependence of variables, providing a way to find a derivative without solving the equation for \(y\).