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Within the world of **calculus**, we explore the** derivative** of **x²** through applications and examples that help us make sense of the myriad phenomena in science and engineering. The **derivative **is a tool that helps us understand **rates of change** and** slopes of curves**. A classic and instructive example is the **derivative** of **x²**, a simple parabolic function.

In this article, we will delve deep into understanding the** derivative** of **x²**, its computation, and the fundamental insights it provides into the function’s behavior. From the realms of pure **mathematics** to **physics** and **engineering**, this **derivative** holds a key place, demonstrating the **quintessential nature** of **calculus** in our comprehension of the **universe**.

### Defining Derivative of x²

The **derivative** of a function quantifies the **rate** at which the output of the function changes with respect to changes in its input. In the context of **x²**, its **derivative** provides the** rate of change** of the **square** of **x** with respect to **x** itself.

Mathematically, the **derivative** of a function **f(x)** at a specific point **x** is defined as the limit as Δ**x** approaches **0** of the **difference quotient** [**f(x + Δx) – f(x)**]/Δ**x**. Applying this to the function **f(x) = x²**, we find that the **derivative**, often denoted as **f'(x)** or **df(x)/dx**, equals **2x**.

As a result, any point **x** on the curve will be true. **y = x²**, the **rate of change** at that point is **2x**. Hence, the **derivative** of the function **x²** gives provides us the slope of the curve’s tangent line **y = x²** at any point (**x, x²**) on the curve.

This result is fundamental in the **calculus** and has significant implications in various fields, such as **physics**, **economics**, and **engineering**, where understanding the **rate of change** of quantities is crucial.

### Graphical Representation of **Derivative **of **x²**

The function **f(x) = x²** is a simple parabolic function, which **graphically** represents a **parabola** opening upwards with its vertex at the origin **(0, 0)**. The result of taking this function’s derivative is **f'(x) = 2x**. Below we present the graphical representation of the function **f(x) = x² **in Figure-1.

Figure-1.

**Graphically**, the function **f'(x) = 2x** is a straight line that passes through the **origin**. The **slope** of this line is **2**, indicating that for each unit increase in **x**, the function value increases by **2 units**. This line cuts the **x-axis** at the origin and divides the plane into **two halves**, with the function being positive in the **right half** (for **x > 0**) and negative in the** left half** (for **x < 0**). Below we present the graphical representation of the function **f'(x) = 2x **in Figure-2.

Figure-2.

Moreover, the function **f'(x) = 2x** represents the angle at which the curve’s tangent line slopes **y = x²** at any point (**x, x²**) on the curve. When **x = 0**, the **derivative** is also **0**, indicating a **horizontal tangent** at the vertex of the **parabola** **y = x²**. As the x-axis is extended away from the origin, the value of the derivative increases or decreases **linearly**.

This corresponds to the** parabola y = x²** getting **steeper** as we move away from the **vertex** in either direction and the angle at which the tangent line to the curve slopes matches the value of the **derivative** at that point.

**Properties**

The **derivative** of the function **f(x) = x²** is **f'(x) = 2x**, and it possesses several key properties that emerge from the fundamental principles of **calculus**.

**Linearity**

This is a **critical property** of all **derivatives**, not just the derivative of **x²**. It indicates that the **derivative** of a constant times a function is the same as the **derivative** of the constant times the function, and the derivative of a constant times the product of two functions equals the total of the **derivatives** of the two functions. If we consider a function **g(x) = ax² + bx** (where **a** and **b** are constants), its derivative would be **g'(x) = 2ax + b**, demonstrating the linearity property.

**Increasing Function**

The **derivative** **f'(x) = 2x** is an** increasing** function. This means that as **x** increases, the value of **2x** also increases. Therefore, the slope of the **tangent line** to the curve **y = x²** increases as we move from left to right along the curve. This reflects the fundamental property of the **parabola y = x²**, which gets** steeper** as we move away from its vertex.

**Slope of Tangent**

The **derivative** of **x²** at a given point provides the slope of the **tangent to the curve** **y = x²** at that point. For example, if we take **x = 3**, then the derivative **f'(3) = 2*3 = 6**. This reveals that the point’s **tangent line’s slope **to the curve **(3, 9)** is **6**.

**Instantaneous Rate of Change**

The **derivative** **f'(x) = 2x** represents the instantaneous rate of change of **y = x²** with respect to **x**. That is, it shows how rapidly the square of a number changes as the number itself changes.

**Null at Origin**

The **derivative** of **x²** is zero when **x = 0**, meaning there is a **horizontal tangent** to the curve **y = x²** at the origin. This corresponds to the fact that the function **x²** reaches a **minimum** value at **x = 0**.

**Symmetry**

The **derivative** **f'(x) = 2x** is a **symmetric function **with respect to the origin since it is an odd function. This **aligns** with the fact that the function **x²** and its **derivative** share the same **symmetry axis**, the y-axis.

By understanding these properties, one gains a deeper comprehension of the **derivative** of **x²** and how it reflects the characteristics of the function it is derived from. This understanding is also fundamental to applying **calculus** in solving **real-world problems**.

**Applications **

The **derivative** of the function **x²** plays a crucial role in several fields, often where the concept of change, growth, or rates is essential. Below, we’ve highlighted its applications in a few different areas:

**Physics**

In **physics**, the derivative of **x²** frequently arises when dealing with **motion**. A function of time can frequently be used to represent the position of an item travelling down a line. If an **object’s location** is indicated by **s(t) = t²**, its **velocity**, which is the derivative of the position function, is given by **v(t) = 2t**. This tells us how fast the object is moving at any instant.

**Economics**

In **economics**, derivatives are used to model **cost functions**. As an illustration, if the whole cost of production **x** units is given by **C(x) = x²**, the derivative, **C'(x) = 2x**, indicates the cost of producing one additional unit, or the marginal cost. This information is invaluable in deciding production levels to **maximize** profits.

**Engineering**

In various branches of **engineering**, the **derivative** of **x²** has applications in **optimization problems**, **control systems**, and **modeling physical systems**. For example, if the signal strength of a **transmitter** varies as the square of the distance from it, understanding the **rate of change** of signal strength can be crucial in designing **efficient communication systems**.

**Computer Graphics**

In **computer graphics**, the derivative of curves, like the **parabola** **x²**, is used for **rendering** and **animation**. By understanding how the curve changes at each point (its derivative), **graphics software** can create smooth and realistic representations of **objects** and **motion**.

**Biology**

In **biology**, the **derivative** of **x²** can be used in population models where a **population’s growth rate** is **proportional** to the size of the population itself.

**Environmental Science**

In **environmental science**, such concepts may be used in **pollutant spread** or **heat distribution models**, where rates of change are crucial for understanding and predicting **outcomes**.

In all these fields, the fundamental idea is the same: the **derivative** of a function, including **x²**, gives us an understanding of how a **quantity** changes in response to changes in input. This is a powerful concept with broad applicability across disciplines.

**Exercise **

**Example 1**

What is the **tangent line’s slope** to the curve,** y = x²** at the point **(2,4)**?

**Solution**

To determine the slope of the **curve’s tangent line** at a specific location, we take the derivative of the function and evaluate it at the given x-coordinate. The derivative of y = x² is:

y’ = 2x

To find the slope at point (2,4), we substitute x = 2 into the derivative, yielding:

y'(2) = 2 * 2

y'(2) = 4

Consequently, the angle between the tangent line to the curve and the point **(2,4)** is **4**. Below we present the same in graphical form.

Figure-3.

**Example 2**

At what points on the curve **y = x²** does the **tangent line** pass through the origin?

**Solution**

A line that passes through the origin has the equation **y = mx**, where **m **is the slope of the line. If the tangent line to the curve **y = x²** passes through the origin, its slope at the point** (x, x²)** must be **x** because the line connects (x, x²) and (0, 0). Therefore, we set the derivative equal to x:

2x = x

Solving this equation gives us **x = 0**, indicating that the only point on the curve **y = x²** where the tangent line passes through the origin is at **(0,0).**

**Example 3**

What is the **tangent line’s slope** to the curve, **y = x²** at the point **(3, 9)**?

**Solution**

To determine the slope of the **curve’s tangent line** at a specific location, we first find the function’s derivative to determine the tangent line’s slope. The derivative of y = x² is:

y’ = 2x

The slope of the tangent line at x = 3 is thus:

y'(3) = 2 * 3

y'(3) = 6

A line with slope m passing through a point (x₁, y₁) has the equation y – y₁ = m(x – x₁). Substituting m = 6 and (x₁, y₁) = (3, 9) gives us:

y – 9 = 6(x – 3)

or equivalently:

y = 6x – 9

Below we present the same in graphical form.

Figure-4.

**Example 4**

Suppose a **particle** is moving along a line such that its position at any time** t** (in seconds) is given by **s(t) = t²** (in meters).What is the particle’s **speed** at?** t = 3 seconds**?

**Solution**

Here, the particle’s velocity is the derivative of the position function. The derivative of **s(t) = t²** is:

s'(t) = 2t

So, the velocity at **t = 3** is:

s'(3) = 2*3

s'(3) = 6 meters per second

**Example 5**

Suppose a company’s** total cost** **C** (in dollars) of producing **x** units of a product is given by **C(x) = 500x²**. What is the **marginal cost** when **x = 100**?

**Solution**

The marginal cost is the rate of change of the total cost with respect to the number of units produced, i.e., it’s the derivative of the cost function. The derivative of C(x) = 500x² is:

C'(x) = 1000x

Therefore, the marginal cost at** x = 100** is:

C'(100) = 1000*100

C'(100) = $100,000 per unit

*All images were created with MATLAB.*