(Q4.) So, you think you can take the derivative?

TL;DR

Learn how to find the first derivative of a product of three functions by using logarithmic differentiation.

Transcript

okay for question number four our equation is y is equal to e to the x times x minus 1 squared times x plus 2 to the fourth power it's a product of three functions and we are trying to find the first derivative of this and whenever you have a product of like more than two functions and especially you have the exponents here this is what you should ... Read More

Key Insights

• 🥡 Taking the natural logarithm of a product of functions can simplify differentiation by transforming it into a sum of logarithms.
• 😀 The derivative of ln(y) can be expressed as 1/y * y', where y' represents the first derivative of the original equation.
• 🤝 The chain rule and logarithmic properties are essential for finding the derivative when dealing with exponents and logarithms.
• 😑 Careful algebraic manipulation and simplification are necessary to obtain the correct derivative expression.

Questions & Answers

Q: What is the purpose of taking the natural logarithm of both sides in this problem?

Taking the natural logarithm allows us to transform the product of three functions into a sum of three logarithms, simplifying the differentiation process.

Q: How does ln(e^x) simplify in the derivation process?

ln(e^x) simplifies to just x, since ln and e^x are inverse functions.

Q: What property of logarithms allows the exponent in ln(x+2)^4 to be brought to the front?

The property is that the exponent can be brought to the front of the logarithm, resulting in 4ln(x+2), without subtracting 1, as this is not a power rule question.

Q: How is the equation simplified after combining the terms with common denominators?

After combining the terms with common denominators, the equation becomes y' = e^x * (x-1) + 2 / (x-1) + 4 / (x+2), with some terms simplified further.

Summary & Key Takeaways

• The equation y = e^x * (x-1)^2 * (x+2)^4 is a product of three functions, and the goal is to find the first derivative.

• By applying logarithmic differentiation, ln y can be taken on both sides, transforming the product into a sum of three logarithms.

• After algebraic manipulations, the derivative can be obtained by applying the chain rule and basic logarithmic properties.