Office_Shredder said:I'm confused. When you say you can't assume f(x)=ln(x), then how is f(x) defined?
Office_Shredder said:Ok. Can you calculate f'(x) from that definition?
Office_Shredder said:That's correct, except it's 1/x not 1/t
yossup said:ah right. i know it's intuitive that 1/x is infinitely differentiable but what do we use to rigorously prove it? taylor series?
Office_Shredder said:Well, can you calculate the derivative of 1/x? How about the second derivative of 1/x? The third derivative?
Once you see a pattern, think about a proof by induction
A smooth function is one that is infinitely differentiable, meaning that it has derivatives of all orders at every point in its domain. This means that the function is continuous and has a well-defined slope at every point, without any sudden changes or breaks in its graph.
Proving that ln(x) is a smooth function is important because it is a fundamental function in mathematics and many scientific fields such as physics, engineering, and economics. It is also a key component in the study of calculus and differential equations.
To prove that ln(x) is a smooth function, we can use the definition of a derivative and apply the limit definition to show that the function has derivatives of all orders at every point in its domain. We can also use the properties of logarithms and the chain rule to simplify the process.
The domain of ln(x) is all positive real numbers, or x > 0. This is because the natural logarithm is only defined for positive numbers, as the inverse of the exponential function y=e^x which has a range of y > 0.
The smoothness of ln(x) means that its graph is continuous and has a well-defined slope at every point. This results in a smooth, curved graph that increases slowly as x increases. The graph also has a vertical asymptote at x=0, where the function is undefined.