What is the derivative of ln(x) and how does it relate to the graph of 1/x?

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In summary, the derivative of ln(x) is equal to 1/x, even for negative values of x, due to the chain rule. The graph of ln(x) only has valid x values to the right of the x axis, while the graph of 1/x has valid values for all real numbers except 0. This shows that functions and their derivatives do not always occupy the same domain, and the values to the left of the y-axis should not be ignored.
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NekotoKoara
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I feel like I might be missing something here, but how is the deriviative of ln(x) equal to 1/x? The graph of ln(x) only has valid x values to the right of the x axis. The graph for 1/x has valid values for x for all real numbers except for 0. Am I to deduce that functions and their derivatives do not always necessarily occupy the same domain? If so are the values to the left of the y-axis of any significance or should they just be ignored?
 
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For ##x < 0## we get with the chain rule ##\frac{d}{dx}\log (-x) = \frac{1}{-x}\cdot (-1) = \frac{1}{x}## and both give ##\frac{d}{dx}\log |x| = \frac{1}{x}## for ##x \neq 0## which is the proper formula for the derivative of the logarithm function for both branches of ##\frac{1}{x}##.
 
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FAQ: What is the derivative of ln(x) and how does it relate to the graph of 1/x?

1. What is the derivative of ln(x)?

The derivative of ln(x) is 1/x.

2. How do you find the derivative of ln(x)?

To find the derivative of ln(x), you can use the power rule, which states that the derivative of x^n is n*x^(n-1). In the case of ln(x), n=1, so the derivative is 1*x^(1-1) = 1*x^0 = 1. Therefore, the derivative of ln(x) is 1/x.

3. Is the derivative of ln(x) the same as the derivative of log(x)?

Yes, the derivative of ln(x) is the same as the derivative of log(x), as both are logarithmic functions with a base of e. The notation ln(x) is often used in calculus, while log(x) is more commonly used in other fields of mathematics.

4. What is the graph of the derivative of ln(x)?

The graph of the derivative of ln(x) is a hyperbola with a horizontal asymptote at y=0. The derivative is positive for all values of x greater than 0, and negative for all values of x less than 0. This reflects the fact that the slope of ln(x) is positive for x>1, negative for 0

5. Why is the derivative of ln(x) equal to 1/x?

The derivative of ln(x) is equal to 1/x because the natural logarithm function is the inverse of the exponential function, e^x. By definition, the derivative of e^x is e^x, and the inverse relationship between these two functions results in the derivative of ln(x) being 1/x.

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