# Nakafa Framework: LLM

URL: /en/subject/high-school/12/mathematics/derivative-function/application-of-derivative
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/12/mathematics/derivative-function/application-of-derivative/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Application of Derivatives",
  description: "Learn how derivatives solve real-world physics problems. Calculate velocity, acceleration, and maximum heights with step-by-step examples and formulas.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/26/2025",
  subject: "Derivative Functions",
};

## Understanding Motion Through Derivatives

Derivatives are not just an abstract concept in mathematics; they are a powerful tool for understanding **change**. One of their most tangible applications is in physics, especially for analyzing the motion of an object. If we know the position function of an object at any given time, we can use derivatives to find its velocity and even its acceleration at any moment.

## Instantaneous Velocity from Position

Imagine you are driving a car. Your position is constantly changing over time. **Velocity** is the rate of change of that position. If we have a function that describes position, <InlineMath math="s(t)" />, then the instantaneous velocity, <InlineMath math="v(t)" />, at time <InlineMath math="t" /> is the first derivative of the position function.

<BlockMath math="v(t) = s'(t) = \frac{ds}{dt}" />

This means we can determine the precise velocity at any instant, not just the average velocity over a journey.

### Determining the Velocity Function

For example, the motion of a particle is determined by the position function <InlineMath math="s(t) = t^3 - 6t^2 + 9t" />, where <InlineMath math="s" /> is in meters and <InlineMath math="t" /> is in seconds. To find its velocity function, we simply differentiate the function <InlineMath math="s(t)" /> with respect to <InlineMath math="t" />.

<div className="flex flex-col gap-4">
  <BlockMath math="v(t) = \frac{d}{dt}(t^3 - 6t^2 + 9t)" />
  <BlockMath math="v(t) = 3t^2 - 12t + 9" />
</div>

With this function, we can calculate the particle's velocity at any time. For instance, at <InlineMath math="t = 1" /> second, the velocity is <InlineMath math="v(1) = 3(1)^2 - 12(1) + 9 = 0" /> meters/second. This zero result indicates that the particle is momentarily at rest, perhaps pausing for a moment or even changing direction.

## Acceleration from Velocity

Now, what if the velocity itself is changing? Perhaps you're pressing the gas pedal or the brakes. This change in velocity is called **acceleration**. Just as velocity is the derivative of position, acceleration, <InlineMath math="a(t)" />, is the first derivative of the velocity function. This also makes it the second derivative of the position function.

<BlockMath math="a(t) = v'(t) = s''(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}" />

### Determining the Acceleration Function

Continuing with the previous particle example, we already have the velocity function <InlineMath math="v(t) = 3t^2 - 12t + 9" />. We can find its acceleration function by differentiating the function <InlineMath math="v(t)" />.

<div className="flex flex-col gap-4">
  <BlockMath math="a(t) = \frac{d}{dt}(3t^2 - 12t + 9)" />
  <BlockMath math="a(t) = 6t - 12" />
</div>

From this, we can find the particle's acceleration at any time. For example, at <InlineMath math="t = 3" /> seconds, its acceleration is <InlineMath math="a(3) = 6(3) - 12 = 6" /> meters/second².

## Peak Application in Vertical Motion

A ball is thrown straight up. Its height, <InlineMath math="h(t)" /> in meters, after <InlineMath math="t" /> seconds is given by the equation <InlineMath math="h(t) = 50t - 5t^2" />.

First, we can find its velocity and acceleration functions.

Velocity function:

<BlockMath math="v(t) = h'(t) = 50 - 10t" />

Acceleration function:

<BlockMath math="a(t) = v'(t) = -10" />

> The acceleration here is constant and negative. This represents Earth's **gravitational acceleration**, which always pulls objects downward. Its value is approximately <InlineMath math="-9.8" /> m/s², but it is often rounded to <InlineMath math="-10" /> in physics calculations to simplify things.

One of the most interesting questions is: **when does the ball reach its maximum height?**

The ball reaches its highest point exactly when it stops for a moment to change direction and fall back down. In other words, this occurs when its velocity is exactly zero.

<div className="flex flex-col gap-4">
    <BlockMath math="v(t) = 0" />
    <BlockMath math="50 - 10t = 0" />
    <BlockMath math="10t = 50" />
    <BlockMath math="t = 5" />
</div>

So, the ball reaches its peak at 5 seconds. To find out what the maximum height is, we substitute <InlineMath math="t=5" /> back into the initial height function.

<BlockMath math="h(5) = 50(5) - 5(5)^2 = 250 - 125 = 125" />

Thus, the maximum height the ball reaches is **125 meters**. Through derivatives, we not only describe motion but can also analyze and determine crucial moments, such as the highest point of a projectile.