Derivatives: The Derivative Function

Derivatives: The Derivative Function

In calculus, the derivative of a function is a fundamental concept that measures how a function's output changes as its input changes. It tells us about the rate of change of a function at any given point. The derivative function refers to the function that gives the derivative of the original function at each point in its domain.

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1. What is the Derivative?

The derivative of a function $f(x)$ at a specific point $x = c$ represents the instantaneous rate of change of the function at that point. It is the slope of the tangent line to the curve at $x = c$, and it tells us how quickly the function is changing at that point.

Derivative Notation:

The derivative of a function $f(x)$ is commonly denoted as:

$$f'(x)$$

or

$$\frac{d}{dx}[f(x)]$$

These notations represent the rate of change of $f(x)$ with respect to $x$.

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2. The Derivative Function

The derivative function is a function that gives the derivative of $f(x)$ at any point $x$ in its domain. In other words, if $f(x)$ is a function, the derivative function $f'(x)$ describes how $f(x)$ changes as $x$ changes for all values of $x$.

Definition of the Derivative Function:

The derivative of $f(x)$, denoted by $f'(x)$, is defined as the limit of the average rate of change of the function as the interval approaches zero:

$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

Where:

• $f(x)$ is the original function.
• $h$ is a small change in $x$.
• $f'(x)$ gives the instantaneous rate of change of $f(x)$ at any point $x$.

This definition is also known as the difference quotient.

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3. How to Find the Derivative Function

To find the derivative function $f'(x)$, you need to apply a set of rules (called derivative rules) to $f(x)$. These rules include:

• Power Rule
• Sum Rule
• Product Rule
• Quotient Rule
• Chain Rule

Each of these rules helps you differentiate different types of functions.

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4. Derivative Rules

Let's briefly look at some common derivative rules:

Power Rule:

If $f(x) = x^n$, where $n$ is any real number, the derivative is:

$$f'(x) = nx^{n-1}$$

Sum Rule:

If $f(x) = g(x) + h(x)$, then the derivative of the sum is the sum of the derivatives:

$$f'(x) = g'(x) + h'(x)$$

Product Rule:

If $f(x) = g(x) \cdot h(x)$, then the derivative of the product is:

$$f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)$$

Quotient Rule:

If $f(x) = \frac{g(x)}{h(x)}$, then the derivative of the quotient is:

$$f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{(h(x))^2}$$

Chain Rule:

If $f(x) = g(h(x))$, where $g$ is a function of $h(x)$, the derivative of $f(x)$ is:

$$f'(x) = g'(h(x)) \cdot h'(x)$$

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5. Example: Finding the Derivative Function

Let's walk through an example to find the derivative function of a simple function.

Example 1: Find the Derivative of $f(x) = 3x^2 + 5x$

We want to find $f'(x)$.

1. Apply the Power Rule: For each term of the function, we use the power rule.

• For $3x^2$, the derivative is $2 \times 3x^{2-1} = 6x$.
• For $5x$, the derivative is $1 \times 5x^{1-1} = 5$.

2. Combine the results:

$$f'(x) = 6x + 5$$

Thus, the derivative function is:

$$f'(x) = 6x + 5$$

This function $f'(x)$ gives the rate of change of $f(x) = 3x^2 + 5x$ at any point $x$.

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Example 2: Find the Derivative of $f(x) = \frac{x^2 + 1}{x}$

1. Simplify the Function: First, rewrite $f(x)$ to make it easier to differentiate:

$$f(x) = \frac{x^2 + 1}{x} = x + \frac{1}{x}$$

2. Apply the Power Rule: Now, differentiate each term.

• The derivative of $x$ is 1.
• The derivative of $\frac{1}{x} = x^{-1}$ is $-x^{-2} = -\frac{1}{x^2}$.

3. Combine the results:

$$f'(x) = 1 - \frac{1}{x^2}$$

Thus, the derivative function is:

$$f'(x) = 1 - \frac{1}{x^2}$$

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6. Geometric Interpretation of the Derivative Function

The derivative function $f'(x)$ gives the slope of the tangent line to the curve $y = f(x)$ at any point $x$. The slope of the tangent line represents the instantaneous rate of change of $f(x)$ at that point.

• At $x = c$, $f'(c)$ gives the slope of the tangent line at $x = c$.
• If $f'(x) > 0$, the function is increasing at that point.
• If $f'(x) < 0$, the function is decreasing at that point.
• If $f'(x) = 0$, the function has a horizontal tangent line, which may indicate a local maximum or minimum (a critical point).

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7. Summary

• The derivative of a function $f(x)$ is a measure of how $f(x)$ changes as $x$ changes.
• The derivative function $f'(x)$ is the function that gives the rate of change of $f(x)$ at each point in its domain.
• To find the derivative function, we use derivative rules such as the power rule, sum rule, product rule, quotient rule, and chain rule.
• The derivative can be interpreted geometrically as the slope of the tangent line to the graph of the function at any given point.

By finding the derivative function, we can better understand the behavior of the function and analyze how it changes at each point.