How to Find dy/dx for y sinh^-1(x/2)

In this article, we will explore the process of finding the derivative dy/dx for the function y sinh^-1(x/2). This involves understanding hyperbolic functions and their derivatives, specifically the inverse hyperbolic sine function. Let's break down the steps and derive the solution in detail.

Understanding Hyperbolic Functions

To begin, it's essential to familiarize ourselves with the definitions and properties of hyperbolic functions. The hyperbolic sine (sinh) and hyperbolic cosine (cosh) functions are defined as follows: sinhx (e^x - e^{-x})/2 coshx (e^x e^{-x})/2 Additionally, the fundamental identity for hyperbolic functions is given by cosh^2x - sinh^2x 1.

Given the Problem and Initial Equations

We are given the function y sinh^{-1}(x/2). To find its derivative, we first express it in a more convenient form. Recall that the inverse hyperbolic sine function is defined as: sinhy x/2 Starting from this, we can rewrite the equation as follows: sinhy x/2 Using the definition of sinh, we substitute and solve for x: (e^y - e^{-y})/2 x/2 Simplifying, we get: e^y - e^{-y} x

Applying Implicit Differentiation

Next, we apply implicit differentiation on both sides of the equation. This means taking the derivative of both sides with respect to x. Let's differentiate each term carefully: e^y dy - e^{-y} - 1 dy dx Simplifying the left-hand side, we get: e^y dy - e^{-y} - dy dx Since dy is common in the first and third terms, we can factor it out: e^y dy - e^{-y} - dy dx Rewriting, we have: e^y dy - (e^{-y} 1) dy dx Dividing both sides by e^y - e^{-y}, we get: dy (dx)/(e^y e^{-y}) Using the definition of cosh, we substitute: dy (dx)/(2cosh y) Since cosh y (e^y e^{-y})/2, we can rewrite the equation as: dy (dx)/(2cosh y)

Substituting Back the Hyperbolic Identity

Now, we use the hyperbolic identity cosh^2y - sinh^2y 1 to substitute back for cosh y. We know that sinh y x/2, so we can substitute sinh y into the identity: cosh^2y - (x/2)^2 1 Solving for cosh y, we get: cosh y sqrt(1 (x/2)^2) Substituting this back into our derivative expression, we obtain: dy/dx 1/(2 * sqrt(1 (x/2)^2)) Simplifying further, we get: dy/dx 1/(2 * sqrt(4x^2/4)) Simplifying inside the square root, we find: dy/dx 1/(2 * (2| x |)) Finally, we simplify the expression to find the derivative dy/dx: dy/dx 1/(4| x |)

Final Expression

The final expression for the derivative dy/dx of the function y sinh^-1(x/2) is: dy/dx 1/(4| x |)

Conclusion

In conclusion, by using the definitions and properties of hyperbolic functions, along with implicit differentiation, we were able to find the derivative dy/dx for the given function y sinh^-1(x/2). This process underscores the importance of understanding hyperbolic functions and their derivatives in solving more complex mathematical problems. Practice and familiarity with these concepts are key to mastering such derivations.

Keywords: hyperbolic functions, inverse hyperbolic sine, derivatives