Derivative of an Integral (Fundamental Theorem of Calculus)
Common Types of Mistakes
The fundamental theorem of calculus provides a formula for finding the derivative of a definite integral when the upper limit of the integral is the variable of differentiation:
We list some common types of mistakes students make when finding such a derivative. Compare the form of each mistake type with the correct formula above.
Type 1 (Wrong variable in the answer)
Example:
The integral defines a function of the variable x, so the answer should be a function of x. Further, when a definite integral is computed, the variable of integration does not appear in the answer. Here's the mistaken formula that was used:
Type 2 (Differentiating under the integral sign)
Example:
The theorem gives no direction to differentiate the integrand (technical issues concerning the differentiability of f aside). Unless f(a) = 0, the following calculation shows that differentiating under the integral sign gives a different answer than the correct formula for the fundamental theorem of calculus:
(A further issue that often arises is that the derivative of the integrand often gets computed incorrectly, further dooming the calculation.)
Type 3 (Attempting to evaluate without using the theorem)
The beauty of the fundamental theorem of calculus is that the derivative of an integral with the upper limit the variable of differentiation can be computed without ever finding an antiderivative. In particular, the conclusion holds even if there is no elementary function antiderivative for the integrand. The mistakes made in this category are generally incorrect antiderivatives that are attempted as part of a direct brute-force approach (that ignores using the fundamental theorem). Often in these problems there is no nice antiderivative, so the direct approach is bound to fail.
Type 4 (Failing to reverse the limits when required)
If the variable of integration appears as the lower limit of integration, then the limits must be reversed first (with a consequent change in sign of the integral) before the fundamental theorem of calculus can be applied. Visit when the lower limit of the integral is the variable of differentiation.
Type 5 (Failing to use the chain rule (at all, or correctly) when required)
If either (or both) limits of the integral involves a function of the variable of differentiation, then the chain rule must be used along with the fundamental theorem of calculus. Visit when one limit or the other is a function of the variable of differentiation and when both limits involve the variable of differentiation.
Other types of mistakes occur, but these are the ones that are made most often.
These derivatives are very easy if you stick to using the exact form of the formula. Don't mess around with "creative" alternative techniques!
More:
- Why must we use two different variables?
- Variations
- Common mistakes on derivative of integral problems
- The Common Calculus Mistakes section of mathmistakes.info includes mistakes on derivatives of integrals.
