Laplace Transform of the first derivative, Laplace transform of f'(t)

TL;DR

This video explains how to find the Laplace transform of the derivative of a function using integration by parts.

Transcript

okay suppose that we know the laas transform of the original function f of T is equal to F of s and we're assuming that the laas transform of f of T it does exist that's trying to find out what's the laas transform of its derivative right and as you can see we don't have a specific function here so we are going to come with a formula based off whic... Read More

Key Insights

• 🥳 The Laplace transform of the derivative can be found using integration by parts.
• 🍉 The formula for the Laplace transform of the derivative involves terms related to the original function and its derivative.
• 🪈 The approach relies on the exponential order property of the original function.
• 🥳 Integration by parts is a useful technique for solving various mathematical problems.

Questions & Answers

Q: What is the purpose of finding the Laplace transform of the derivative?

Finding the Laplace transform of the derivative is useful in solving differential equations and analyzing dynamic systems. It allows us to transform derivative equations into algebraic equations that are simpler to solve.

Q: How is the formula for the Laplace transform of the derivative derived using integration by parts?

The formula is derived by choosing the derivative of the exponential term as the "di" part and the integral of the original function as the "i" part. This leads to the product of the exponential term and the original function as the first part of the answer.

Q: Why is integration by parts used instead of other methods?

Integration by parts is chosen because it allows us to differentiate the exponential term, which leads to cancellation of terms and simplification in the integration process. Other methods, such as u-substitution, are not as effective in this case.

Q: What does it mean for a function to have exponential order?

A function is said to have exponential order if it grows or decays exponentially as T approaches infinity. This property is important for ensuring that the integral involving the Laplace transform converges.

Summary & Key Takeaways

• The video discusses finding the Laplace transform of the derivative of a function when the Laplace transform of the original function is known.

• Integration by parts is used to derive the formula for finding the Laplace transform of the derivative.

• The integration by parts method involves choosing one part of the integral to differentiate and the other part to integrate.

• The final result for the Laplace transform of the derivative involves terms related to the original function and its derivative.