How to Find the Laplace Transform of Derivatives

TL;DR

To find the Laplace transform of the first derivative of a function, use the formula L{f'(t)} = sL{f(t)} - f(0), where L denotes the Laplace transform. For the second derivative, the formula is L{f''(t)} = s²L{f(t)} - sf(0) - f'(0). This approach simplifies solving differential equations.

Transcript

the goal for this video is to figure out what's a Laplace transform of F prime of T and here let's assume that F prime T in original function f of T they are both of exponential order so that will make sure that we will be able to give a lot of last transform of these functions so this is something that we're trying to do and we hope to reduce this... Read More

Key Insights

• 🇹🇫 The Laplace transform of F prime of T is obtained by integrating the product of e to the negative s T and f prime of T.
• 🥳 Integration by parts is used to evaluate the integral, resulting in a formula that relates the Laplace transform of the derivative to the Laplace transform of the original function and the value of the original function at 0.
• 😑 By applying this formula, the Laplace transform of the second derivative of a function can be expressed in terms of the Laplace transform of the first derivative and the Laplace transform of the original function.
• 🇹🇫 Assuming both F prime of T and f of T are of exponential order ensures that the Laplace transform of these functions exists and can be computed accurately.
• 🆓 The Laplace transform of F prime of T can be used to solve differential equations efficiently.
• 🎮 The formula provided in this video is a useful tool for solving differential equations involving Laplace transforms.
• 🔨 The Laplace transform is a powerful mathematical tool for converting differential equations into algebraic equations.

Questions & Answers

Q: What is the Laplace transform of F prime of T?

The Laplace transform of F prime of T is given by the formula s times the Laplace transform of the original function minus the value of the original function at 0.

Q: How is the Laplace transform of a derivative related to the Laplace transform of the original function?

The Laplace transform of a derivative is equal to s times the Laplace transform of the original function minus the value of the original function at 0.

Q: What is the significance of assuming both F prime of T and f of T are of exponential order?

Assuming both F prime of T and f of T are of exponential order ensures that the Laplace transform of these functions exists and can be evaluated.

Q: What is the purpose of using integration by parts in finding the Laplace transform of F prime of T?

Integration by parts is used to evaluate the integral in the Laplace transform of F prime of T, as it involves a product of two functions. This technique allows us to simplify the integral and obtain a useful formula.

Summary & Key Takeaways

• The goal of this video is to determine the Laplace transform of F prime of T, assuming F prime T and the original function f of T are both of exponential order.

• The Laplace transform of F prime of T is defined as an integral from 0 to infinity of e to the negative s T times f prime of T, where the integration is done with respect to T.

• Integration by parts is used to evaluate this integral, and the result is a formula that relates the Laplace transform of the derivative to the Laplace transform of the original function and the value of the original function at 0.