Derivative of an Integral (Fundamental Theorem of Calculus)
The fundamental theorem of calculus may be used in combination with other derivative rules. One example will show the flavor of such calculations.
The ex factor in the integrand in independent of the variable of integration t, and therefore by the constant multiple rule for integrals can be pulled in front of the integral. The derivative is then taken using the product rule, using the fundamental theorem of calculus to differentiate the integral factor (in this case, using the chain rule as well):
While the answer may be unsatisfying in that it involves the initial integral, it does show that the function y(x) defined by the integral
is a solution of the differential equation
As the example shows, the technique of finding the derivative of an integral (with the variable of integration involved in one or both limits of integration) is just another technique that can be used in combination with the other techniques of differentiation.
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