Define a function f : R → R byf (x) =sin(x2)x if x̸ = 0,0 if x = 0.Determine whether f is differentiable at 0 or not. If f is differentiable at 0, then compute f ′(0)

To determine if the function f is differentiable at 0, we need to check if the limit of the difference quotient exists at that point.

The difference quotient is given by:

(f(x) - f(0)) / (x - 0)

For x ≠ 0, f(x) = sin(x^2) / x and f(0) = 0. So, the difference quotient becomes:

(sin(x^2) / x - 0) / x = sin(x^2) / x^2

We need to find the limit of this expression as x approaches 0.

We can use the squeeze theorem here. We know that -1 ≤ sin(x^2) ≤ 1 for all x. So, -1/x^2 ≤ sin(x^2) / x^2 ≤ 1/x^2. As x approaches 0, -1/x^2 and 1/x^2 both approach infinity. Therefore, by the squeeze theorem, the limit of sin(x^2) / x^2 as x approaches 0 is 0.

Since the limit of the difference quotient exists, the function f is differentiable at 0.

To find f'(0), we need to compute the derivative of f(x) = sin(x^2) / x for x ≠ 0 and then evaluate it at x = 0.

The derivative of f(x) can be found using the quotient rule:

f'(x) = (x * cos(x^2) * 2x - sin(x^2)) / x^2

Evaluating this at x = 0 gives f'(0) = 0.

So, the function f is differentiable at 0 and f'(0) = 0.

To determine if the function f is differentiable at 0, we need to check if the limit of the difference quotient exists at that point.

The difference quotient is given by:

f'(x) = lim (h->0) [f(x+h) - f(x)] / h

For x ≠ 0, the derivative of f(x) = sin(x^2) / x can be found using the quotient rule for differentiation, which states that the derivative of two functions u(x) and v(x) is given by:

(u/v)' = (v * u' - u * v') / v^2

Let u(x) = sin(x^2) and v(x) = x. Then u'(x) = 2x cos(x^2) and v'(x) = 1.

So, f'(x) = (x * 2x cos(x^2) - sin(x^2) * 1) / x^2 = 2 cos(x^2) - sin(x^2) / x.

For x = 0, f(x) = 0, so f'(0) = lim (h->0) [f(h) - f(0)] / h = lim (h->0) sin(h^2) / h.

As h approaches 0, sin(h^2) also approaches 0, and the limit of sin(h^2) / h as h approaches 0 is 0.

Therefore, f is differentiable at 0, and f'(0) = 0.