Derivation of the Inverse Hyperbolic Sine Function sinh^(-1)(x) = ln(x + sqrt(x^2 + 1))

TL;DR

Learn to find the inverse of the hyperbolic sine function step by step using exponential manipulation.

Transcript

everyone this is kind of an interesting problem so we have the hyperbolic sine function and we're going to find the inverse from scratch we're just going to do it let's try it so solution so to find the inverse of the function usually you start by calling the function what so we're going to Y be equal to the hyperbolic sine of X is H X now what is ... Read More

Key Insights

• 👨‍💼 The hyperbolic sine function is defined as half the difference of e to the X and e to the negative X.
• 😫 Switching the X and Y variables helps set up the equation to find the inverse function.
• 👨‍💼 Exponential manipulation and the quadratic formula are essential tools for solving the inverse of the hyperbolic sine function.

Questions & Answers

Q: What is the first step in finding the inverse of the hyperbolic sine function?

The first step involves defining the hyperbolic sine function as half the difference of e to the X and e to the negative X.

Q: How do you set up the equation for finding the inverse of the hyperbolic sine function?

By switching X and Y in the function, you can establish the equation to be solved to find the inverse.

Q: What manipulation technique is used to solve for the inverse of the hyperbolic sine function?

Exponential manipulation is utilized to convert the function into a quadratic equation in order to apply the quadratic formula for solving it.

Q: What is the final result of finding the inverse of the hyperbolic sine function?

The inverse of the hyperbolic sine function is obtained as y = natural log(X + sqrt(x^2 + 1)), representing the completed inverse function.

Summary & Key Takeaways

• Define the hyperbolic sine function as half the difference of e to the X and e to the negative X.

• Switch X and Y to set up the equation for finding the inverse.

• Use exponential manipulation and the quadratic formula to solve for the inverse function.