Consider for example, the function y = 3x² + 2x - 1, then dy/dx = 6x + 2, which is reasonable.

When x < -1/3, dy/dx or tanθ is negative and when x > -1/3, dy/dx or tanα is positive. When x= 1/3, the slope of the tangent line is horizontal and thus equal to 0. These correspond to its derivative function figure and it is obvious that the function is differentable at all points.

A function is differentiable at a point if the derivative of the function exists at that point. A function is differentiable on an interval if it is differentiable at every point in the interval, as already concluded.

According to the above definition, y = 3x² + 2x - 1 function is differentiable.

y = (x²)^0.5

(x²)^0.5 is not
Differentiable at all Points

Now, consider another example, y = (x²)^0.5. Is y differentiable in this case at all points?

The definition of derivative gives

     

In order to applying the definition to the y = (x²)^0.5 function, Δy/Δx need to be find first.

At x = 0,

The left hand limit is:

The right hand limit is:

The left hand limit is not equal to the right hand limit. Thus does not exist, in other words, the function y = (x²)^0.5 is not differentiable when x = 0.

If a function is differentiable at point b, then this function is continuous at this specific point.

proof:

When x approaches b, the difference between f(x) and f(b) is y_x - y_b. Since x ≠ b, y_x - y_b can be written as:

Thus

Differentiability and Continuity I

Differentiability and Continuity II

Therefore,

This proves that when y is differentiable at point b, it is continuous at that point.

Function y = 3x² + 2x - 1 is differentiable in its variable range, and so it is continuous.

If a function is continuous, is it differentiable? The answer is may not be. For example y = (x²)^0.5 is continuous, but it was shown that it is not differentiable.

Function y = (x²)^0.5 is not differentiable at 0 because its left hand limit is not equal to its right hand limit. Normally, if a function's graphic has a corner or kink (loop), then the function is not differentiable.

When the curve of a continuous function has a vertical tangent line at point a, the slope of the tangent line is infinity. This means

Thus the function is not differentiable.