How do you take the derivative of inverse sine?

Unlocking the Secrets of Inverse Sine: A Calculus Adventure

Okay, inverse sine. Arcsin(x) or sin⁻¹(x) – whatever you call it, it can look a bit scary, right? But trust me, cracking its derivative is totally doable, and it unlocks some seriously cool stuff in engineering, physics, even how your GPS works! So, let’s break it down, step by step.

What’s the Deal with Inverse Sine, Anyway?

Before we get all deriv-y (yeah, I just made that up), let’s make sure we’re on the same page. You know how the regular sine function takes an angle and spits out a number between -1 and 1? Well, arcsin basically reverses that. You give it a number between -1 and 1, and it tells you what angle has that sine. Think of it like asking, “Hey, what angle gives me this sine value?”.

The Magic Formula: Derivative of Arcsin

Alright, drumroll please… The derivative of arcsin(x) is:

1 / √(1 – x²)

Boom! That’s it. Simple, right? This little formula tells you how fast the arcsin(x) function is changing at any given point. But where does it come from? Let’s find out.

How We Get There: The Implicit Differentiation Tango

This is where the fun begins. We’re going to use a technique called implicit differentiation to prove the formula. It’s like a mathematical dance, so follow along:

Ta-da! We just proved the formula. Feels good, doesn’t it?

Arcsin in Action: Real-World Examples

Okay, enough theory. Let’s see this in action:

Example 1: What if you have f(x) = arcsin(3x)?

• Chain rule to the rescue! Let u = 3x. Then f(x) = arcsin(u).
• d/dx (arcsin(3x)) = 1/√(1 – (3x)²) * d/dx (3x) = 1/√(1 – 9x²) * 3 = 3/√(1 – 9x²)

Example 2: Imagine you’re designing a bridge, and its curve follows the equation y = arcsin(x). You need to know how steep the bridge is at x = 1/2 meters.

• dy/dx = 1/√(1 – x²)
• Plug in x = 1/2: dy/dx = 1/√(1 – (1/2)²) = 1/√(3/4) = 2/√3

So, at x = 1/2 meters, the bridge’s elevation is changing at a rate of 2/√3.

Watch Out for These Traps!

• arcsin(x) ISN’T 1/sin(x): This is a classic mistake. Arcsin is the inverse, not the reciprocal. Big difference!
• Don’t forget the chain rule: If you’re taking the derivative of arcsin(something more complicated than just ‘x’, like arcsin(u)), remember to multiply by the derivative of that “something” (du/dx).
• Mind the domain: Arcsin only plays nice with numbers between -1 and 1. Don’t try plugging in anything else!

Final Thoughts

The derivative of arcsin(x) might seem abstract, but it’s a powerful tool. Once you understand where it comes from and how to use it, you’ll be surprised how often it pops up. So, go forth, practice, and conquer those inverse trigonometric functions!