i got 2 Functions Y(x)=lxl^3 f(x)=x-1
1) Prove that Y can be Differentiable In point 0
2) use the information from ur Proof In 1 to Prove that the Function l(x-1)l^3
are Differentiable In point 1
thanks for helping.
$\displaystyle \lim_{h\rightarrow 0}\frac{Y(0+h)-Y(0)}{h}=\lim_{h\rightarrow 0}\frac{|h|^3}{h}$.
Now compute the limit from the left and the limit from the right separately. They both give 0, showing that the derivative of Y at 0 exists and is equal to 0.
For the second one use the derivative theorem about the composition of 2 functions.
If you need clarification on the right and left hand limits let me know.
Yes. If h is positive, then |h| is just h. If h is negative, then |h| is -h.
So if h is positive $\displaystyle \frac{|h|^3}{h} = \frac{h^3}{h} = h^2$. So the limit from the right is $\displaystyle 0^2 = 0$.
If h is negative, $\displaystyle \frac{|h|^3}{h} = \frac{(-h)^3}{h} = -h^2$. So the limit from the left is $\displaystyle -0^2 = 0$.