Question

First find and simplify $$\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ Then find \(d y / d x\) by taking the limit of your answer as \(\Delta x \rightarrow 0 .\) $$ y=x^{2} $$

Short answer

The derivative \( \frac{dy}{dx} \) of \( y = x^2 \) is \( 2x \).

Step-by-step solution

Define the Function and Setup the Expression

We start with the given function \( y = x^2 \). We need to find \( \Delta y / \Delta x \) using the expression \( \frac{f(x+\Delta x) - f(x)}{\Delta x} \). Substituting \( f(x) = x^2 \), we have \( f(x+\Delta x) = (x+\Delta x)^2 \) and \( f(x) = x^2 \).

Expand the Binomial

Expand \( f(x+\Delta x) = (x+\Delta x)^2 \) using the binomial theorem: \( (x+\Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2 \).

Substitute into the Difference Quotient

Substitute the expanded form back into the difference quotient: \[ \frac{(x+\Delta x)^2 - x^2}{\Delta x} = \frac{x^2 + 2x\Delta x + (\Delta x)^2 - x^2}{\Delta x} \].

Simplify the Difference Quotient

Simplify the expression by canceling \( x^2 \) terms: \[ \frac{2x\Delta x + (\Delta x)^2}{\Delta x} \]. Factor \( \Delta x \) out from the numerator: \[ \frac{\Delta x(2x + \Delta x)}{\Delta x} \]. Cancel \( \Delta x \) from the numerator and denominator to get: \[ 2x + \Delta x \].

Take the Limit as \(\Delta x \to 0\)

Now, take the limit of the simplified expression as \( \Delta x \to 0 \): \[ \lim_{\Delta x \to 0} (2x + \Delta x) = 2x \]. Thus, the derivative \( \frac{dy}{dx} \) of \( y = x^2 \) is \( 2x \).

Key concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus that helps us find the derivative of a function. Imagine you have a curve, and you're interested in finding the slope of the tangent line at a certain point. The difference quotient provides a way to approximate this slope by considering the slope of a secant line between two points on the curve.

In mathematical terms, if you have a function \( f(x) \), the difference quotient is expressed as:

• \( \frac{f(x+\Delta x) - f(x)}{\Delta x} \)

This formula represents the average rate of change of the function over the interval \( \Delta x \). As \( \Delta x \) becomes very small, the secant line approaches the tangent line, providing us with the instantaneous rate of change.
The simplified form of this quotient allows us to find the derivative, the key tool for understanding how functions change at any instant.

Limit

The concept of a limit is crucial in understanding the process of finding derivatives through the difference quotient. A limit essentially tells us what happens to a function's value as its input approaches a particular point.

For differentiation, we're interested in what happens as \( \Delta x \) approaches zero. This concept is expressed mathematically by writing:

• \( \lim_{\Delta x \to 0} (\text{expression}) \)

Taking the limit allows us to transition from the average rate of change, given by the difference quotient, to the exact slope of the tangent line at a single point.
When solving for the derivative of \( y = x^2 \), we found the simplified difference quotient to be \( 2x + \Delta x \). By taking the limit as \( \Delta x \to 0 \), we conclude that the slope, or the derivative, is \( 2x \).

Binomial Expansion

Binomial expansion is a mathematical method that simplifies expressions raised to a power, often making them easier to work with. When finding derivatives, this technique is handy for expanding expressions such as \((x + \Delta x)^2\).

Using the binomial expansion, we can break down \((x + \Delta x)^2\) into:

• \( x^2 + 2x\Delta x + (\Delta x)^2 \)

This expansion is part of the process of substituting into the difference quotient. Breaking down complicated expressions into simpler terms makes it easier to perform further simplifications, leading us to cancel terms effectively and find the derivative more smoothly.
Binomial expansion thus plays a pivotal role in deriving the simplified form of the difference quotient.

Function Differentiation

Function differentiation involves finding the derivative, which reveals how a function changes at any given point. By determining the derivative, we gain insights into properties like the rate of change, slopes of curves, and the behaviors of various functions.

The differentiation process generally involves several steps:

• Identifying the function to differentiate.
• Applying the difference quotient to express the rate of change over an interval.
• Simplifying this expression using algebraic techniques like binomial expansion.
• Taking the limit as the interval \( \Delta x \) becomes infinitely small to find the exact rate of change, or slope, at a specific point.

For our function \( y = x^2 \), function differentiation took us through these steps to arrive at the derivative \( 2x \). This tells us that the slope of the function at any point \( x \) on the curve is proportional to \( x \) itself, demonstrating how each concept interlinks within the broader tapestry of calculus.