How to Find the Derivative of f(x) = 1/(2x)

TL;DR

To find the derivative of f(x) = 1/(2x) using the limit definition, you calculate the limit as h approaches 0 of [f(x+h) - f(x)]/h. This process involves algebraic manipulation leading to the result, which is -1/(2x^2), upon evaluating the limit.

Transcript

hello in this video we're going to find the derivative of f of x equals 1 over 2x using the definition of the derivative let's go ahead and work through it very carefully solution so the definition of the derivative is the following so F Prime of x is equal to the Limit as H approaches 0 of f of x plus h minus f of x and all of this is being divide... Read More

Key Insights

• ⛔ The definition of the derivative involves evaluating the limit of a difference quotient.
• ☺️ Substitution is used to find the function at x + h and simplify the expression.
• 🍉 Cancelling out terms and carefully evaluating the limit help derive the final derivative.
• ⛔ Understanding the concept of limit approaching 0 is crucial in finding the derivative accurately.
• ❓ The process of finding a derivative involves algebraic manipulations and arithmetic calculations.
• 🫥 Tangent lines represent instantaneous rates of change on a function curve.
• ❓ Simplifying fractions and recognizing patterns assist in deriving the derivative efficiently.

Questions & Answers

Q: How is the derivative of a function calculated using the definition of derivative?

The derivative of a function is found by evaluating the limit of a difference quotient as h approaches 0, which represents the slope of the tangent line at a specific point on the function curve.

Q: What is the significance of the term h approaching 0 in the derivative calculation process?

As h approaches 0, the secant line between two points on the function curve becomes a tangent line at a single point, allowing for the calculation of the instantaneous rate of change (derivative) at that point.

Q: How is the substitution done when finding the derivative of the given function?

Substitution involves replacing x in the function f(x) with x + h to obtain f(x + h), which is then used in the difference quotient formula to calculate the derivative.

Q: Why is it essential to simplify each step when calculating the derivative?

Simplifying each step ensures clarity and accuracy in the calculation process, making it easier to identify and cancel out terms to reach the final derivative expression.

Summary & Key Takeaways

• The video demonstrates finding the derivative of f(x) = 1/(2x) using the definition of derivative.

• The process involves substitution, simplification, and evaluating the limit as h approaches 0.

• By carefully following the steps, the final answer is derived as -1/(2x^2).