## When does a derivative not exist at ‘x’ (with a graph)?

### A derivative does not exist at a certain point ‘x’ on a graph when the tangent line at that point does not have a well-defined slope

A derivative does not exist at a certain point ‘x’ on a graph when the tangent line at that point does not have a well-defined slope. In other words, if the function represented by the graph has a sharp corner, a vertical tangent line, or a discontinuity at ‘x’, then the derivative is undefined at that point.

1. Sharp Corner: If the graph of a function has a sharp turn or corner at ‘x’, the derivative will not exist at that point. At such a corner, the tangent line would need to have two different slopes (one on each side of the corner), which is not possible.

2. Vertical Tangent Line: If the graph has a vertical tangent line at ‘x’, it means that the slope of the tangent line approaches infinity or negative infinity. In this case, the derivative is undefined at that point because it does not have a finite value.

3. Discontinuity: If the graph has a discontinuity at ‘x’ (either a jump, removable, or essential discontinuity), the derivative does not exist at that point. A jump discontinuity occurs when there is a sudden change in the graph, causing a jump in the function values. A removable discontinuity happens when there is a hole in the graph that can be removed by defining the function value at that point. An essential discontinuity is when there is an infinite oscillation or an asymptote near ‘x’.

It is important to note that for a function to be differentiable at a point, it must be continuous at that point. Differentiability implies continuity, but continuity does not guarantee differentiability.

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