# taking a derivative that WolframAlpha can't handle | Summary and Q&A

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February 9, 2022
by
blackpenredpen
taking a derivative that WolframAlpha can't handle

## TL;DR

This video explains how to differentiate integral functions using the product rule and the fundamental theorem of calculus.

## Key Insights

• 🎮 The video demonstrates the differentiating process for the product of two integral functions using the product rule and substitution.
• ❓ The fundamental theorem of calculus is a crucial concept for differentiating integral functions.
• 🎮 The video emphasizes the importance of understanding the chain rule and substitution when differentiating integral functions.
• ❓ It is not necessary to combine fractions or find common denominators when differentiating integral functions.
• 🥡 The process of taking the natural logarithm does not require an absolute value when the argument is always positive.
• 📼 The video mentions the possibility of finding the maximum of a function and hints at the need to set the numerator equal to zero.
• 💬 The video encourages viewers to leave comments suggesting topics for future differentiation examples.

## Transcript

just because i'm a calculus teacher a lot of  people like to ask me when is the best time to   take the derivative well let's take a look right  now it's 3:20 pm and here we go that's the answer   and we will be differentiating the  product of these two integral functions   and as i told you we will make derivatives cool  this year so here we go th... Read More

### Q: How do you differentiate the product of two integral functions?

To differentiate the product of two integral functions, we use the product rule. We keep the first function as it is and differentiate the second function, then add the two results together.

### Q: How do you differentiate an integral function with a variable as an upper limit?

To differentiate an integral function with a variable as an upper limit, we substitute the variable into the integral function and differentiate the result, taking into account the chain rule.

### Q: What is the fundamental theorem of calculus?

The fundamental theorem of calculus states that if F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b can be evaluated by subtracting F(a) from F(b).

### Q: Why don't we need to include the absolute value when taking the natural logarithm?

In this case, we don't need the absolute value because the function inside the natural logarithm is always positive. If there is a possibility of a negative argument, we would need to consider the absolute value.

## Summary & Key Takeaways

• The video demonstrates the process of differentiating the product of two integral functions using the product rule.

• It explains how to differentiate an integral function with a variable as an upper limit by substituting the variable.

• The fundamental theorem of calculus is introduced, and its application in differentiating integral functions is demonstrated.