# What is the derivative of ln(x^2 + y^2)?

• ace123
In summary, the correct answer for finding y' when y=ln(x^2+y^2) using implicit differentiation is (2x+2y)/(x^2+y^2). However, it is important to note that y' can be in terms of both x and y due to the use of implicit differentiation.
ace123
Find y' if y= ln( x^2 + y^2)

I thought this was just a regular natural log derivative combined with the chain rule.
So what I got was (2x + 2y)/ (x^2 + y^2)

But this wasn't the correct answer. So could someone help point out my mistake.
Thank you.

This is implicit differentiation. Since you have y = f(x,y) instead of y = f(x), you have to assume y does equal to some function of x then take derivatives on both sides and solve for y'. Since we made the assumption y can be written as some function of x, D(y^2) = 2yy'. Now, you have to solve for y' just in terms of x and y to get the final answer.

Forgot to mention y' can be in terms of x AND y since we don't know what y is there's no way to remove it from the expression without solving for y in the original equation which is what we were trying to avoid in the first place.

Last edited:
Oh okay thank you

## 1. What is the definition of natural logarithmic function?

The natural logarithmic function is a mathematical function that is the inverse of the exponential function. It is written as ln(x) and is the logarithm to the base e, where e is a mathematical constant approximately equal to 2.71828.

## 2. What is the relationship between natural logarithmic function and exponential function?

The natural logarithmic function and the exponential function are inverse functions of each other. This means that if we take the natural logarithm of a number, the result will be the exponent of the same number. For example, ln(e^x) = x and e^(ln(x)) = x.

## 3. How is natural logarithmic function used in real life?

Natural logarithmic function has many applications in fields such as finance, physics, and biology. It is used to model exponential growth and decay, such as in population growth or radioactive decay. It is also used in calculating interest rates and in the calculation of pH in chemistry.

## 4. What is the domain and range of natural logarithmic function?

The domain of natural logarithmic function is all positive real numbers (x > 0) and the range is all real numbers (y ∈ R). This means that the input (x) must be a positive number, and the output (y) can be any real number.

## 5. How is natural logarithmic function different from common logarithmic function?

The main difference between natural logarithmic function and common logarithmic function is their base. Natural logarithmic function has a base of e, while common logarithmic function has a base of 10. This means that ln(x) = loge(x) and log(x) = log10(x). Natural logarithmic function is also used more frequently in mathematical and scientific calculations compared to common logarithmic function.

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