on April 24, 2022

What is the derivative of inverse sine?

Space & Navigation

Decoding the Arcsin Derivative: It’s Simpler Than You Think!

Ever wondered about the inverse sine function – arcsin(x), or sin⁻¹(x) if you prefer? It’s basically asking, “Hey, what angle gives me a sine of x?” Now, if you’re diving into calculus, knowing its derivative is super useful. Trust me, it pops up everywhere from engineering to physics. So, let’s break down the derivative of arcsin(x) in a way that actually makes sense.

The Big Reveal: What’s the Derivative?

Okay, drumroll please… The derivative of the inverse sine function is:

d/dx (arcsin x) = 1 / √(1 – x²)

That’s it! This formula tells you how quickly the arcsin(x) function changes as x changes. Think of it like this: if you nudge the input x just a tiny bit, how much does the output of the arcsin function wiggle? Just a heads up, this formula works perfectly when x is between -1 and 1. Outside that range, things get a little wonky.

Let’s Prove It (Without the Headache)

There are a few ways to prove this thing – implicit differentiation, the chain rule, even going back to first principles. But let’s stick with implicit differentiation. It’s pretty straightforward and gives you a good “aha!” moment.

  • Start with the Basics:

    Let’s say y = arcsin(x). That’s just another way of saying sin(y) = x. Simple, right?

  • Differentiate Like a Pro (Implicitly):

    Now, let’s take the derivative of both sides of sin(y) = x with respect to x. Remember the chain rule? We get:

    cos(y) * (dy/dx) = 1

    Basically, we’re acknowledging that y is a function of x, so we need to account for that inner derivative.

  • Isolate the Good Stuff (dy/dx):

    Let’s get dy/dx by itself. Just divide both sides by cos(y):

    dy/dx = 1 / cos(y)

  • Time to Get Triggy (Express cos(y) in terms of x):

    We know sin(y) = x. Remember that old Pythagorean identity, sin²(y) + cos²(y) = 1? It’s our ticket! We can rewrite cos(y) as:

    cos(y) = √(1 – sin²(y)) = √(1 – x²)

    Now, here’s a little detail: arcsin(x) lives in the range of -π/2 to π/2. Cosine is always positive in that range, so we can stick with the positive square root.

  • The Grand Finale (Substitute and Celebrate!):

    Plug cos(y) = √(1 – x²) back into dy/dx = 1 / cos(y), and BAM!

    dy/dx = 1 / √(1 – x²)

    And there you have it! The derivative of arcsin(x) is, without a doubt, 1 / √(1 – x²).

    • Real-World Stuff: Applications and Examples

    This derivative isn’t just some abstract math thing. It’s a workhorse in calculus, especially when you’re dealing with integrals or trying to optimize something. For example, it shows up when you’re integrating functions that look like 1/√(a² – x²).

    Quick Example: Let’s find the derivative of f(x) = arcsin(3x).

    Using the chain rule again:

    f'(x) = 1 / √(1 – (3x)²) * d/dx (3x) = 3 / √(1 – 9x²)

    Watch Out for These Traps!

    One common mistake? Thinking arcsin(x) is the same as 1/sin(x). Nope! Arcsin(x) is the inverse sine, not the reciprocal. Also, don’t forget the chain rule when you’re differentiating something like arcsin(3x). It’s easy to slip up!

    Final Thoughts

    The derivative of the inverse sine function – that’s 1 / √(1 – x²) – is a key tool in calculus. Getting comfy with its derivation, especially using implicit differentiation, sets you up for tackling tougher problems. Steer clear of those common mistakes, practice a bit, and you’ll be differentiating arcsin like a seasoned pro!

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