I’ll explain the derivative of arctan(2x) in a simpler way.

Imagine you have a big playground, and there are two friends, let’s call them x and y. They are playing a game where x moves along the horizontal line, and y moves along the vertical line. They have a special rule: y’s movement is determined by x’s movement.

Now, let’s say x moves a little bit to the right, and we want to know how much y moves in response to that. We can measure their movements using a special tool called a derivative.

In this game, the relationship between x and y is given by the arctan(2x) function. It means that for any value of x, we can find the corresponding value of y using this function.

To find out how much y moves when x moves a little bit, we need to find the derivative of the arctan(2x) function.

The derivative tells us the rate at which y is changing with respect to x. In other words, it tells us how fast y is moving compared to x. We use a special notation called dy/dx to represent the derivative.

To find the derivative of arctan(2x), we can use a formula that helps us calculate derivatives of different functions. Applying this formula, we find that the derivative of arctan(2x) is:

dy/dx = 2 / (1 + (2x)^2)

Now, this might look a bit complicated, but what it means is that if x moves a little bit to the right, y will move a certain amount determined by this formula.

So, by using the formula, we can find out exactly how fast y is moving compared to x for any given value of x in the game they are playing.

That’s how we find the derivative of arctan(2x)! It helps us understand how much y moves when x moves in their special game on the playground.

Let’s work through three examples of finding the derivative of the function arctan(2x) and explain the steps involved.

Example 1: Let’s find the derivative of y = arctan(2x).

Step 1: We need to apply the derivative formula, which states that the derivative of arctan(2x) is given by 2 / (1 + (2x)^2).

Step 2: Substitute the expression 2x into the formula: dy/dx = 2 / (1 + (2x)^2) dy/dx = 2 / (1 + 4x^2) dy/dx = 2 / (4x^2 + 1)

So, the derivative of y = arctan(2x) is dy/dx = 2 / (4x^2 + 1).

Example 2: Consider the function y = 3arctan(5x).

Step 1: Apply the derivative formula: dy/dx = 2 / (1 + (2x)^2)

Step 2: Substitute the expression 5x into the formula: dy/dx = 2 / (1 + (2 * 5x)^2) dy/dx = 2 / (1 + 100x^2) dy/dx = 2 / (100x^2 + 1)

Thus, the derivative of y = 3arctan(5x) is dy/dx = 2 / (100x^2 + 1).

Example 3: Let’s find the derivative of y = arctan(x^3).

Step 1: Apply the derivative formula: dy/dx = 2 / (1 + (2x)^2)

Step 2: Substitute the expression x^3 into the formula: dy/dx = 2 / (1 + (2 * x^3)^2) dy/dx = 2 / (1 + 4x^6) dy/dx = 2 / (4x^6 + 1)

Therefore, the derivative of y = arctan(x^3) is dy/dx = 2 / (4x^6 + 1).

In each example, we first applied the derivative formula, and then substituted the appropriate expression for x into the formula. This allowed us to simplify the expression and find the derivative of the given function.