Lebnitz Rule | Differentiation Under Integral Sign
Online Mathematics Tutorial・2 minutes read
The video explains the rule of differential under the integral sign for converting differential equations into integral equations using fundamental calculus theorems. It explores various scenarios showing how the rule varies based on constants and functions, demonstrating its practical application.
Insights
- Understanding the rule of differential under the integral sign is essential for transforming differential equations into integral forms, showcasing the significance of this mathematical concept in problem-solving.
- The proof of this rule relies on the total derivative of functions f(X, B) and f(X, A), demonstrating the intricate relationship between derivatives and integrals in calculus, which is pivotal for grasping the underlying principles of this mathematical technique.
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Recent questions
What is the rule of differential under the integral sign?
The rule of differential under the integral sign is crucial for converting differential equations into integral ones. It involves proving the equation u = ∫a to b f(X, T) DT using fundamental calculus theorem.
How is the rule of differential under the integral sign proven?
The rule is proven by using the derivative of u with respect to X, which involves the total derivative of f(X, B) and f(X, A). This process is essential for demonstrating the validity of the rule.
What does the total derivative of a function entail?
The total derivative of a function involves considering all variables that the function depends on, not just one. It is crucial for understanding how a function changes with respect to all of its variables.
In what scenarios does the rule of differential under the integral sign change?
The rule changes based on the constants or functions involved in the integral equation. Different scenarios are explored to exemplify how the rule adapts to varying conditions.
How is the rule of differential under the integral sign applied in integral equations?
The rule is applied in integral equations to convert differential equations into integral ones. By understanding the rule and its variations, one can effectively manipulate integral equations for solving complex problems.
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Summary
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Rule of Differential Under Integral Sign Explained
- The video discusses the rule of differential under the integral sign, crucial for converting differential equations into integral ones.
- To prove this rule, fundamental calculus theorem is used, leading to the equation u = ∫a to b f(X, T) DT.
- The derivative of u with respect to X involves the total derivative of f(X, B) and f(X, A), crucial for the proof.
- Different scenarios are explored where the rule changes based on the constants or functions involved, exemplifying its application in an integral equation.
