How to differentiate integrals when the upper limit is a function of x
In short: The Fundamental Theorem of Calculus provides a direct way to differentiate integrals with a variable upper limit. This video explains how to extend that method using the chain rule when the upper limit is a more complex function, such as a trigonometric expression.
The standard Fundamental Theorem of Calculus allows for finding the derivative of an integral defined from a constant to a variable x. However, when the upper limit of integration is itself a function of x—for instance, sin(x)—the standard approach is insufficient on its own.
To solve these cases, one must combine the Fundamental Theorem of Calculus with the chain rule. This technique ensures that the derivative accounts for the inner function's rate of change, allowing for the differentiation of more complex integral expressions.
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Sources
- Summary by Intellect · source: Finding derivative with fundamental theorem of calculus: chain rule | AP®︎ Calculus | Khan Academy
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