Other Tricks

The fundamental theorem of calculus may be used in combination with other derivative rules. One example will show the flavor of such calculations.

Example: Find

The e^x 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.

More:

• Why must we use two different variables?
• Other variations
  • When the lower limit of the integral is the variable of differentiation
  • When one limit or the other is a function of the variable of differentiation
  • When both limits involve the variable of differentiation
• Common mistakes on derivative of integral problems
  • Common types of mistakes
  • The Common Calculus Mistakes section of mathmistakes.info includes mistakes on derivatives of integrals.