How to Find the Derivative Using the Limit Definition

TL;DR

To find the derivative of f(x) = sqrt(x + 2) using the limit definition, start by using the difference quotient and rationalize the numerator to avoid division by zero. After simplifying, the limit as h approaches zero gives the derivative, which is 1/(2*sqrt(x + 2)).

Transcript

all right so you have to find the derivative of this function using the definition uh of the derivative so solution so we'll start by looking at the difference quotient which is f ofx plus now here you can use Delta X or H I'll use H minus F ofx all divid H okay so f of x plus h that's just x + H + 2 uh and F ofx is just the < TK of x + 2 this is a... Read More

Key Insights

• 🥡 The process of finding the derivative using the definition involves the difference quotient and taking the limit.
• 🥡 Rationalizing the numerator helps avoid division by zero when taking the limit.
• 😑 The final derivative expression is obtained by simplifying the expression after taking the limit.

Questions & Answers

Q: What is the first step in finding the derivative using the definition?

The first step is to apply the difference quotient, which involves subtracting the function at x from the function at x plus h, all divided by h.

Q: Why can't we directly take the limit as h approaches zero in the original expression?

Directly taking the limit would result in division by zero. To avoid this, we rationalize the numerator by multiplying by the conjugate.

Q: How do we simplify the rationalized numerator?

By using the difference of squares formula, we simplify the rationalized numerator to x + H + 2 - (x + 2) which simplifies to H.

Q: What happens when we take the limit as h approaches zero in the final expression?

Taking the limit gives us 1 over the square root of x + 2 plus the square root of x + 2, which simplifies to 1 over 2 times the square root of x + 2.

Summary & Key Takeaways

• The content explains the process of finding the derivative of a function using the definition.

• The difference quotient is used to simplify the expression.

• Rationalizing the numerator and taking the limit helps obtain the final derivative expression.