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Using the definition of derivative (i.e. using "first principles"), limit laws and standard limits, find the derivatives of the following functions.

y=ln(1+x)

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Sure, let's start with the given function, y = ln(1 + x).

The derivative of a function is a fundamental concept in calculus, which represents an infinitesimal change in the function with respect to its independent variable.

We will use the chain rule to find the derivative of this function. The chain rule is a formula to compute the derivative of a composite function. That is, if we have a function which is composed of two other functions, then the derivative of that function is the derivative of the first function times the derivative of the second function.

In this case, we consider two functions: u = 1 + x and y = ln(u).

1. The first step to find the derivative of y = ln(1 + x) is to differentiate y = ln(u) with respect to u. From the basic rules of differentiation,

The derivative of y = ln(u) is dy/du = 1/u.

2. The next step is to find the derivative of u = 1 + x with respect to x. By definition, the derivative of a constant is always zero while derivative of x is 1. Therefore, du/dx is just equal to 1.

3. Lastly, we use chain rule which states that dy/dx = dy/du * du/dx. Replacing the values that we have found in Step 1 and Step 2, we get the derivative of y = ln(1 + x) as

dy/dx = 1/u * 1 = 1/(1+x).

The derivative of y = ln(1 + x) is 1/(1+x).

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