Differentiating functions: Find the error | Derivative rules | AP Calculus AB | Khan Academy

TL;DR

This video analyzes mistakes commonly made when taking derivatives and provides correct solutions.

Transcript

• [Instructor] What we're gonna do in this video is look at the work of other people as they try to take derivatives and see if their reasoning is correct and if it's not correct, try to identify what they should have done or where their reasoning went wrong. So over here it says Nate tried to find the derivative of X squared plus five X times sine... Read More

Key Insights

• 🥡 Taking the derivative of a product requires the application of the product rule.
• 📏 The chain rule is essential when finding the derivative of a composition of functions.
• 😨 Care must be taken when dealing with transcendental functions like sine or cosine, as their derivative application can be misleading.

Questions & Answers

Q: What was Nate's mistake in finding the derivative of X squared plus five X times sine of X?

Nate's mistake was assuming that the derivative of a product is equal to the product of the derivatives, instead of applying the product rule.

Q: What was Katy's mistake in finding the derivative of two X squared minus four, all to the third power?

Katy's mistake was not properly applying the chain rule when finding the derivative, as she failed to multiply by the derivative of the inner function.

Q: What was Njoman's mistake when finding the derivative of sine of seven X squared plus four X?

Njoman's mistake was incorrectly multiplying the two expressions in parentheses and assuming the cosine function should be applied to the entire expression.

Q: What was Tom's mistake in finding the derivative of the square root of X over X to the fourth?

Tom did not make any mistakes in finding the derivative, but he could have simplified the expression further by recognizing that he could apply the power rule before using the quotient rule.

Summary & Key Takeaways

• The first example shows Nate's mistake of assuming the derivative of a product is the product of the derivatives, instead of applying the product rule.

• Katy's mistake in finding the derivative of a function to the power of 3 is not properly applying the chain rule.

• Njoman's error lies in multiplying expressions by assuming they should be multiplied because of the presence of parentheses, resulting in an incorrect application of the chain rule.

• Tom correctly applies the quotient rule but could have simplified the expression further using the power rule.