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Noismaker
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What are the implications of x=0 in terms of function derivability?
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In summary, the conversation discusses the concept of differentiability and its relation to the function f(x)= x^x. It is suggested that "differentiability" would be a more suitable term than "derivability" to avoid confusion. The approach of removing the absolute value to analyze positive and negative values of x is also mentioned. Differentiation is then applied to both cases, resulting in the conclusion that f(x) is differentiable for all positive and negative x values. However, at x=0, the derivative does not exist.
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HallsofIvy
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I think "differentiability" would be a better word than "derivability" since that might be confused with the ability to "derive" a formula. In any case, the first thing I would do is remove the "absolute value" by looking at x positive or negative separately.
If x is positive, then [tex]f(x)= x^x[/tex] and, taking the logarithm of both sides, log(f(x))= x log(x). Differentiating, [tex]f'(x)/f(x)= log(x)+ 1[/tex] so that [tex]f'(x)= (log(x)+ 1)f(x)= (log(x)+ 1)x^x[/tex]. That exists for all positive x so f(x) is differentiable for all positive x.
If x is negative, make the substitution y= -x. |x|= -x= y so that [tex]x^x= y^{-y}[/tex]. Again take the logarithm of both sides of [tex]f(y)= y^{-y}[/tex], [tex]log(f(y))= -ylog(y)[/tex]. Differentiating, [tex]f'(y)/f(y)= -log(y)- 1[/tex] so that [tex]f'(y)= -f(y)(log(y)+ 1)= -y^y(log(y)+ 1) and, since dy/dx= 0, [tex]f'(x)= (-x)^x(log(|x|)+ 1)[/tex]. That exists for all negative x so f(x) is differentiable for all negative x.
Finally, look at x= 0. The derivative of f(x) at x= 0 is [tex]\lim_{h\to 0} \frac{f(h)- f(0)}{h}= \lim_{h\to 0}\frac{h^h- 1}{h}[/tex]. Does that limit exist?
If x is positive, then [tex]f(x)= x^x[/tex] and, taking the logarithm of both sides, log(f(x))= x log(x). Differentiating, [tex]f'(x)/f(x)= log(x)+ 1[/tex] so that [tex]f'(x)= (log(x)+ 1)f(x)= (log(x)+ 1)x^x[/tex]. That exists for all positive x so f(x) is differentiable for all positive x.
If x is negative, make the substitution y= -x. |x|= -x= y so that [tex]x^x= y^{-y}[/tex]. Again take the logarithm of both sides of [tex]f(y)= y^{-y}[/tex], [tex]log(f(y))= -ylog(y)[/tex]. Differentiating, [tex]f'(y)/f(y)= -log(y)- 1[/tex] so that [tex]f'(y)= -f(y)(log(y)+ 1)= -y^y(log(y)+ 1) and, since dy/dx= 0, [tex]f'(x)= (-x)^x(log(|x|)+ 1)[/tex]. That exists for all negative x so f(x) is differentiable for all negative x.
Finally, look at x= 0. The derivative of f(x) at x= 0 is [tex]\lim_{h\to 0} \frac{f(h)- f(0)}{h}= \lim_{h\to 0}\frac{h^h- 1}{h}[/tex]. Does that limit exist?
- #3
Noismaker
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1) View attachment 6623
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so in x=0 we have a point of non-differentiation, cusp, angular point, or flattened to vertical tangent.
2) View attachment 6624
3) View attachment 6625
so in x=0 we have a point of non-differentiation, cusp, angular point, or flattened to vertical tangent.
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FAQ: What are the implications of x=0 in terms of function derivability?
1. What is the concept of "derivability of function"?
Derivability of function refers to the mathematical concept of finding the rate of change of a function at a specific point. It is a fundamental concept in calculus and is used to understand the behavior of a function over a given interval.
2. How do you determine if a function is differentiable at a certain point?
To determine if a function is differentiable at a certain point, you need to check if the limit of the difference quotient (the change in y divided by the change in x) exists as the change in x approaches 0. If the limit exists, the function is differentiable at that point.
3. What is the difference between differentiability and continuity?
The main difference between differentiability and continuity is that a function can be continuous at a point without being differentiable at that point. Continuity means that the function has no breaks or holes, while differentiability means that the function has a well-defined slope at that point.
4. Can a function be differentiable but not continuous?
No, a function cannot be differentiable but not continuous. Differentiability implies continuity, since a function must be continuous at a point for it to be differentiable at that point.
5. How is the concept of differentiability used in real life?
The concept of differentiability is used in many real-life applications, such as physics, engineering, and economics. It is used to calculate rates of change, determine the maximum and minimum values of a function, and understand the behavior of a system over time. For example, it can be used to calculate the velocity of a moving object, the growth rate of a population, or the rate of change of stock prices.