Derivative of Arcsin

Today, we’re going to dive into the world of derivatives and explore the derivative of the arcsin function, which is the inverse of the sine function. Don’t worry, I’ll guide you through it step by step, and we’ll look at some examples to make sure you’ve got it down.

What is the arcsin function?

First off, let’s remind ourselves what the arcsin function is. The arcsin function, also known as the inverse sine function, is denoted as arcsin(x) or sin^-1(x). It basically answers the question: what angle in radians (between -π/2 and π/2) has a sine of x?

Now, let’s get to the heart of the matter – the derivative of arcsin. To find the derivative of y = arcsin(x), we can use implicit differentiation. Here’s how:

We start by writing the original function as:

sin(y) = x

Now, we differentiate both sides with respect to x. Remember that when we differentiate the left side, we treat y as a function of x (that is, y = y(x)):

d/dx[sin(y)] = d/dx[x]

Using the chain rule on the left side, we get:

cos(y) * dy/dx = 1

Solving for dy/dx, we get:

dy/dx = 1/cos(y)

But what is cos(y) in terms of x? We know from the definition of sine that sin²(y) + cos²(y) = 1. Since sin(y) = x, we have x² + cos²(y) = 1. Therefore, cos(y) = sqrt(1 - x²).

So, our derivative becomes:

dy/dx = 1/sqrt(1 - x²)

And thus, the derivative of arcsin(x) is:

d/dx[arcsin(x)] = 1/sqrt(1 - x²)

Example:

Let’s look at an example to see this in action. What is the derivative of arcsin(2x)?

Using the chain rule, we get:

d/dx[arcsin(2x)] = 1/sqrt(1 - (2x)²) * d/dx[2x]
                     = 2/sqrt(1 - 4x²)

Possible Questions from Learners:

Q1: What is the range of x for which the derivative of arcsin is defined?
A1: The derivative of arcsin is defined for all x in the interval -1 ≤ x ≤ 1, since the square root requires the argument 1 - x² to be non-negative.

Q2: Why do we use sqrt(1 - x²) and not -sqrt(1 - x²) when finding the derivative?
A2: Good question! We use sqrt(1 - x²) because by definition, the range of arcsin is between -π/2 and π/2, where the cosine is always positive.

Q3: Can we use this method to find the derivative of other inverse trigonometric functions?
A3: Absolutely! You can use a similar process of implicit differentiation and trigonometric identities to find the derivatives of other inverse trigonometric functions like arccos and arctan.

I hope this explanation has made the concept of the derivative of arcsin clearer. Remember, practice makes perfect, so try differentiating some functions involving arcsin on your own. If you have any more questions, feel free to ask!

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