Finding the Derivative of a Quadratic Function

Oct 13, 2025 by thesweetdesigns.com 18 views

**Finding the Derivative of a Quadratic Function** =====================================================

Introduction

In this article, we will explore the concept of finding the derivative of a quadratic function using the definition of a derivative. We will use the given function $f(x)=4x^2-10x+8$ to demonstrate the process.

What is a Derivative?

A derivative is a measure of how a function changes as its input changes. It is a fundamental concept in calculus and is used to study the behavior of functions. The derivative of a function $f(x)$ is denoted by $f^{\prime}(x)$ and is defined as:

f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}

Finding the Derivative of a Quadratic Function

To find the derivative of a quadratic function, we can use the definition of a derivative. Let's start by finding the derivative of the given function $f(x)=4x^2-10x+8$ .

Step 1: Write Down the Definition of a Derivative

The definition of a derivative is given by:

f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}

Step 2: Substitute the Function into the Definition

We will substitute the given function $f(x)=4x^2-10x+8$ into the definition of a derivative:

f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{(4(x+h)^2-10(x+h)+8) - (4x^2-10x+8)}{h}

Step 3: Expand the Expression

We will expand the expression $(x+h)^2$ and simplify the expression:

(x+h)^2 = x^2 + 2xh + h^2

Substituting this into the expression, we get:

f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{(4(x^2 + 2xh + h^2)-10(x+h)+8) - (4x^2-10x+8)}{h}

Step 4: Simplify the Expression

We will simplify the expression by combining like terms:

f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{4x^2 + 8xh + 4h^2 - 10x - 10h + 8 - 4x^2 + 10x - 8}{h}

f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{8xh + 4h^2 - 10h}{h}

Step 5: Cancel Out the Common Factor

We will cancel out the common factor $h$ :

f^{\prime}(x) = \lim_{h \rightarrow 0} 8x + 4h - 10

Step 6: Evaluate the Limit

We will evaluate the limit as $h$ approaches 0:

f^{\prime}(x) = 8x + 0 - 10

f^{\prime}(x) = 8x - 10

Conclusion

In this article, we used the definition of a derivative to find the derivative of a quadratic function. We started by writing down the definition of a derivative, substituted the given function into the definition, expanded the expression, simplified the expression, canceled out the common factor, and evaluated the limit. The final answer is $f^{\prime}(x) = 8x - 10$ .

Q&A

Q: What is the definition of a derivative? A: The definition of a derivative is given by $f^{\prime}(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$ .

Q: How do I find the derivative of a quadratic function? A: To find the derivative of a quadratic function, you can use the definition of a derivative. Start by writing down the definition, substitute the function into the definition, expand the expression, simplify the expression, cancel out the common factor, and evaluate the limit.

Q: What is the derivative of the function $f(x)=4x^2-10x+8$ ? A: The derivative of the function $f(x)=4x^2-10x+8$ is $f^{\prime}(x) = 8x - 10$ .

Q: Why do we need to cancel out the common factor $h$ ? A: We need to cancel out the common factor $h$ to simplify the expression and evaluate the limit.

Q: What is the final answer? A: The final answer is $f^{\prime}(x) = 8x - 10$ .