00:01
In this problem, we have been given a function, f of x, and we need to use the definition of derivative to determine whether f of x is differentiable at x is equal to 0.
00:11
So, using the definition of derivative, if we want to determine whether the function is differentiable, what we need to do is find the left -hand derivative and the right -hand derivative, and we need to show that these two are equal for it to be differentiable.
00:26
So let us consider the left -hand derivative at zero.
00:32
So this will be the limit as h tends to 0 minus of f of 0 plus h minus f of 0 divided by h.
00:45
So what do we end up with? so over here what do we have? we have the limit as h tends to 0 minus f of h minus f of h minus f of 0, divided by h.
00:59
So the limit as h tends to zero minus.
01:02
So h tends to zero minus, that means h approaches zero from the left, and so h is less than zero.
01:08
And whenever x is less than zero, f of x is five x squared, so f of h will be five h squared.
01:16
And here we have f of zero.
01:18
Now, if x is greater than or equal to zero, f of x is three x, so here x is equal to zero, so we have three times 0 and we divide this by h.
01:28
So we end up with the limit as h tends to 0 minus 5h squared minus 0, that's just 58 squared divided by h.
01:37
Now we can reduce that fraction by h because h tends to 0 minus and so it is not equal to 0.
01:44
So we have 58 squared divided by 8.
01:46
That's just 5h.
01:47
And now we can use the direct substitution property.
01:51
So it's 5 times 0.
01:52
That will be 0...