How do you differentiate sin Inverse 2x?

Unlocking the Mystery: Differentiating arcsin(2x) Like a Pro

Okay, so you’re staring at arcsin(2x), also known as sin⁻¹(2x), and wondering how to find its derivative. It can look intimidating, I get it! But trust me, with a little calculus know-how, especially the chain rule, it’s totally doable. Let’s break it down together, step by step.

First, a Quick Refresher

Before we dive in, let’s make sure we’re all on the same page with a few key ideas.

• Inverse Sine, Explained Simply: Think of arcsin(x) as the “undo” button for the sine function. If sin(y) gives you x, then arcsin(x) gets you back to y. Just remember, arcsin(x) only works for x values between -1 and 1, and the result (y) will always be between -π/2 and π/2.
• Derivatives: The Rate of Change: A derivative basically tells you how much a function is changing at any given point. Imagine you’re driving a car; the derivative is like your speedometer, showing how your speed changes as you press the gas pedal.
• The Chain Rule: Your Secret Weapon: This is the big one! The chain rule is how we handle functions inside other functions, like our arcsin(2x). It says: if you have y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). In plain English, you differentiate the outer function first, keeping the inner function as is, and then multiply by the derivative of the inner function. Sounds complicated, but it’s not so bad once you get the hang of it.

The Foundation: Derivative of arcsin(x)

You absolutely need to know this: the derivative of plain old arcsin(x) is 1 / √(1 – x²). Memorize it, tattoo it on your arm, whatever works! This is our starting point.

Cracking the Code: Differentiating arcsin(2x)

Alright, let’s get our hands dirty with arcsin(2x). This is where the chain rule shines.

Boom! The derivative of arcsin(2x) is 2 / √(1 – 4x²). Just a little heads-up: this is only true when x is between -1/2 and 1/2. Otherwise, that square root gets a little…unhappy (we end up with imaginary numbers, which is a whole other can of worms).

Another Way to Skin the Cat: Implicit Differentiation

Want to see another trick? You can also use implicit differentiation. It’s like solving a mystery from a different angle.

Ta-da! Same answer as before. Pretty cool, huh?

Final Thoughts

Differentiating arcsin(2x) might have seemed like a Herculean task, but now you’ve got it down. Whether you prefer the chain rule or implicit differentiation, you’ve added another tool to your calculus toolbox. Keep practicing, and you’ll be differentiating like a math ninja in no time!