Find the Derivative of f(x) = ln(x)/x^3 using the Quotient Rule for Derivatives

TL;DR

This video explains how to find the derivative of the natural logarithm of x divided by x cubed using the quotient rule.

Transcript

in this problem we have to find the derivative of the natural log of x divided by x cubed so we have a quotient so one strategy is to use something called the quotient rule so the quotient rule says if you have two functions say f and g and you're taking the derivative of the quotient so you're dividing them think of f as your top function and g as... Read More

Key Insights

• 📏 The quotient rule is a useful technique for finding the derivative of a quotient of two functions.
• ☺️ The derivative of the natural logarithm of x is simply 1/x.
• ☺️ The power rule can be used to find the derivative of x cubed, resulting in 3x squared.

Questions & Answers

Q: What is the quotient rule?

The quotient rule is a differentiation technique used when finding the derivative of a quotient of two functions. It states that the derivative of the quotient is equal to (derivative of the top function multiplied by the bottom) minus (top function multiplied by the derivative of the bottom), divided by the bottom squared.

Q: How do you find the derivative of the natural logarithm of x?

The derivative of the natural logarithm of x is equal to 1 divided by x. This follows from the derivative of ln(x) being 1/x.

Q: What is the derivative of x cubed?

The derivative of x cubed can be found using the power rule, which states that the derivative of x to the power of n is equal to n times x to the power of (n-1). Applying this rule, the derivative of x cubed is 3 times x squared.

Q: How can the derivative expression be simplified?

To simplify the derivative expression, you can factor out the greatest common factor in the numerator, which is x squared. This will result in x squared multiplied by (1 - 3ln(x)), all divided by x to the power of 6.

Summary & Key Takeaways

• The video discusses the use of the quotient rule to find the derivative of the natural logarithm of x divided by x cubed.

• The quotient rule states that the derivative of the top function multiplied by the bottom, minus the top multiplied by the derivative of the bottom, all divided by the bottom squared.

• The process involves finding the derivatives of the top and bottom functions, applying the quotient rule, simplifying the expression, and obtaining the final answer.