For this post, I wanted to upload an interesting proof of the Laplace transform of Derivatives that is used for transforming a given differential equation in the unknown function into an algebraic equation in . The **Theorem** is stated as follows: Suppose that the function is continuous and piecewise smooth for and is of exponential order as , so that there exist nonnegative constants , , and such that for . Then exists for , and Now for the proof of the **Theorem** in the general case in which is merely piecewise continuous. To do this it must be proven that the limit exists and we also need to find its value. With fixed, let be the points interior to the interval at which is discontinuous. Let and . Then the interval can be integrated by parts on each interval where is continuous. This yields and is equivalent to , which I will denote as equation . Now the first summation in equation is equal to and telescopes down to . The second summation adds up to times the integral from to . Therefore equation reduces to From the **Theorem **above I obtain the expression where if . Therefore, finally taking the limits as in the preceding equation, we get the desired result

# Proof of the Transform of Derivatives

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