Derivative of cosh^-1(x), two ways

TL;DR

Learn how to differentiate the inverse hyperbolic cosine function using two methods for simplification.

Transcript

okay this video will see how to differentiate the inverse hyperbolic cosine function of course you can see the answer on the shirt right here but of course we'll do it legitimately right here i'll show you guys two ways to do it first way is that we can actually just differentiate just like last time we actually have an expression for the inverse c... Read More

Key Insights

• 😑 Directly differentiating the expression involving natural logs simplifies to 1 / sqrt(x^2 - 1).
• ❓ Implicit differentiation is applied when finding the derivative of the inverse hyperbolic cosine function using the original cosh function.
• 😑 The identity between sinh and cosh functions is essential for simplifying the derivative expression.
• ☺️ Simplifying the derivative in terms of x enables a clearer understanding of the relationship between the inverse hyperbolic cosine function and its derivative.
• 🤩 Combining fractions and simplifying expressions are key steps in the differentiation process.
• 🍉 Understanding the chain rule and how it applies to functions with multiple terms is crucial for differentiation.
• 🆘 Utilizing mathematical identities can help simplify complex calculus problems.

Questions & Answers

Q: How can the inverse hyperbolic cosine function be differentiated using the expression involving natural logs?

To differentiate the inverse hyperbolic cosine function using natural logs, you first find the expression in terms of ln(x + sqrt(x^2 - 1)). By applying the chain rule and simplifying, the derivative simplifies to 1 / sqrt(x^2 - 1).

Q: What is the second method to differentiate the inverse hyperbolic cosine function?

The second method involves taking the original cosh function to find the derivative using implicit differentiation. By differentiating cos(y)=x with respect to x, and using the derivative of the cosh function, you can find the derivative as 1 / sinh(y).

Q: How is the identity between sinh and cosh functions used in the differentiation process?

The identity cosh^2(t) - sinh^2(t) = 1 is utilized to connect sinh and cosh functions. By rearranging the identity and taking the positive square root, the relation between sinh and cosh can be used to simplify the derivative expression.

Q: Why is it important to simplify the derivative expression in terms of x for the inverse hyperbolic cosine function differentiation?

Simplifying the derivative expression in terms of x is crucial as it allows for a functional understanding of the relationship between the inverse hyperbolic cosine function and its derivative. This simplification aids in further mathematical analysis and applications.

Summary & Key Takeaways

• Differentiation of the inverse hyperbolic cosine function can be done using the expression in terms of natural logs.

• The first method involves differentiating the expression directly and simplifying the result by combining fractions.

• The second method involves taking the original cosh function to find the derivative using implicit differentiation.