Let $f(x)=\sqrt{2x+10}$. According To The Definition, Find $f^{\prime}(x$\]:$f^{\prime}(x)=\lim _{h \rightarrow 0} \square$ (Enter The Limit In Reduced Form)

**Let $f(x)=\sqrt{2x+10}$. According to the definition, find $f^{\prime}(x)$**

Discussion Category: Mathematics

Introduction The concept of a derivative is a fundamental idea in calculus, and it plays a crucial role in understanding the behavior of functions. In this article, we will explore the definition of a derivative and use it to find the derivative of the function $f(x)=\sqrt{2x+10}$.

What is a Derivative? A derivative is a measure of how a function changes as its input changes. It is defined as the limit of the difference quotient as the change in the input approaches zero. Mathematically, it can be represented as:

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

Finding the Derivative of $f(x)=\sqrt{2x+10}$ To find the derivative of $f(x)=\sqrt{2x+10}$, we will use the definition of a derivative. We will start by finding the difference quotient and then take the limit as $h$ approaches zero.

Step 1: Find the Difference Quotient The difference quotient is given by:

$$\frac{f(x+h)-f(x)}{h}$$

We can substitute the function $f(x)=\sqrt{2x+10}$ into the difference quotient:

$$\frac{\sqrt{2(x+h)+10}-\sqrt{2x+10}}{h}$$

Step 2: Simplify the Difference Quotient To simplify the difference quotient, we can multiply the numerator and denominator by the conjugate of the numerator:

$$\frac{\sqrt{2(x+h)+10}-\sqrt{2x+10}}{h} \cdot \frac{\sqrt{2(x+h)+10}+\sqrt{2x+10}}{\sqrt{2(x+h)+10}+\sqrt{2x+10}}$$

This simplifies to:

$$\frac{(2(x+h)+10)-(2x+10)}{h(\sqrt{2(x+h)+10}+\sqrt{2x+10})}$$

Step 3: Simplify the Expression We can simplify the expression further by combining like terms:

$$\frac{2x+2h+10-2x-10}{h(\sqrt{2(x+h)+10}+\sqrt{2x+10})}$$

This simplifies to:

$$\frac{2h}{h(\sqrt{2(x+h)+10}+\sqrt{2x+10})}$$

Step 4: Take the Limit Now that we have simplified the expression, we can take the limit as $h$ approaches zero:

$$\lim _{h \rightarrow 0} \frac{2h}{h(\sqrt{2(x+h)+10}+\sqrt{2x+10})}$$

This simplifies to:

$$\lim _{h \rightarrow 0} \frac{2}{\sqrt{2(x+h)+10}+\sqrt{2x+10}}$$

Final Answer The final answer is:

$$f^{\prime}(x)=\frac{1}{\sqrt{2x+10}}$$

Q&A

Q: What is the definition of a derivative? A: The derivative of a function $f(x)$ is defined as the limit of the difference quotient as the change in the input approaches zero.

Q: How do you find the derivative of a function? A: To find the derivative of a function, you can use the definition of a derivative and take the limit of the difference quotient as the change in the input approaches zero.

Q: What is the difference quotient? A: The difference quotient is a measure of the change in the function over a small change in the input.

Q: How do you simplify the difference quotient? A: You can simplify the difference quotient by multiplying the numerator and denominator by the conjugate of the numerator.

Q: What is the final answer for the derivative of $f(x)=\sqrt{2x+10}$? A: The final answer is $f^{\prime}(x)=\frac{1}{\sqrt{2x+10}}$.

Conclusion In this article, we used the definition of a derivative to find the derivative of the function $f(x)=\sqrt{2x+10}$. We simplified the difference quotient and took the limit as $h$ approaches zero to find the final answer. The derivative of $f(x)=\sqrt{2x+10}$ is $f^{\prime}(x)=\frac{1}{\sqrt{2x+10}}$.