# Solving Complex Log Derivatives: y = log_2(x^2+1)

• iamsmooth
In summary, the pattern for finding the derivative of a logarithmic function is d/dx[log_b(x)] = 1/(xln(b)). Using this pattern, the derivative of y = log_2(x^2 + 1) is y' = (2x)/(x^2+1)ln(2). This approach may be considered quick and dirty, but it is a valid method for finding the derivative. However, it is important to note that ln(x) is not the same as log(x), so the correct use of natural logarithms should be used in the derivative equation.
iamsmooth

## Homework Statement

$$y = \log_{2}(x^2 + 1)$$

## Homework Equations

I think the pattern is:

$$\frac{d}{dx}[\log_{b}(x)] = \frac{1}{x ln(b)}$$

## The Attempt at a Solution

$$y\prime = \frac{2x}{(x^2+1)ln(2)}$$

I did this by applying the pattern (that may or may not be correct) and then chain ruling the middle. If this is correct, then would this amount of work be acceptable (as you can kind of eye it without doing much work)?

When we do weird functions like $$y=x^x^2$$ I know how to do them by taking the ln of both sides and playing around with log properties, since this is the only kind of question that came up on quizzes, it's the only kind of log derivatives I'm familiar with.

Anyways, thanks.

Is there a reason why the denominator log(2) instead of ln(2)?

ln(x) != log(x), no?

Thanks for the webpage, seems awesomely useful for future reference.

First line below the derivative states "log(x) is the natural logarithm"...

Oh whoops, sorry.

Thanks a lot, appreciate the timely help :D

## 1. How do I solve for the derivative of log_2(x^2+1)?

To solve for the derivative of log_2(x^2+1), we can use the chain rule and the power rule. First, rewrite the equation as y = (x^2+1)^(1/2). Then, take the derivative of the inside function, which is 2x, and multiply it by the derivative of the outside function, which is (1/2)(x^2+1)^(-1/2). This gives us the final derivative of y = (x^2+1)^(1/2) * 2x/(x^2+1).

## 2. Can I use the quotient rule to solve for the derivative of log_2(x^2+1)?

No, the quotient rule is used for finding the derivative of a function that is in the form of f(x)/g(x). In this case, the function is log_2(x^2+1), which cannot be rewritten as a quotient of two functions.

## 3. What is the domain of the function y = log_2(x^2+1)?

The domain of the function y = log_2(x^2+1) is all real numbers greater than or equal to 0. This is because the argument of the logarithm function (x^2+1) must be positive, and the base of the logarithm function (2) must be positive as well.

## 4. How can I graph the function y = log_2(x^2+1)?

To graph the function y = log_2(x^2+1), you can plot points by choosing different values for x and calculating the corresponding y values. Alternatively, you can use a graphing calculator or software to graph the function. The resulting graph will be a curve that approaches the x-axis as x approaches negative or positive infinity, and has a vertical asymptote at x = 0.

## 5. Is there a way to simplify the expression log_2(x^2+1)?

No, the expression log_2(x^2+1) cannot be simplified further. However, you can rewrite it as (1/2)log_2(x^2+1)^2, using the power rule for logarithms. This may make it easier to take the derivative, but it is not considered a simplified form of the expression.

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