Of sin, cos, or tan, the graph that is not differentiable at all points is tan (tangent).

Tan(x) is defined as sin(x)/cos(x), and it exhibits vertical asymptotes at points where cos(x) = 0. At these points, the derivative of tan(x) is undefined, making it not differentiable. The vertical asymptotes occur at x = π/2, 3π/2, 5π/2, etc. These points correspond to the peaks and troughs in the tan graph, where the slope of the curve becomes infinitely steep.

On the other hand, sin(x) and cos(x) are differentiable at all points. The derivatives of sin(x) and cos(x) are cos(x) and -sin(x), respectively, which are continuous functions. This means that sin(x) and cos(x) have well-defined slopes at every point on their graphs.

When it comes to differentiability, it is important to consider the behavior of a function at specific points. A function is said to be differentiable at a point if it has a unique tangent line at that point. If the function has a sharp turn or a vertical tangent at a point, it is not differentiable at that point.

In the case of tan(x), the vertical asymptotes create sharp turns in the graph, leading to points where the function is not differentiable. This is in contrast to sin(x) and cos(x), which have smooth and continuous graphs without any sudden changes in slope.

Overall, while sin(x) and cos(x) are differentiable at all points, tan(x) is not differentiable at points where cos(x) = 0. Understanding the differentiability of trigonometric functions is essential in calculus and other advanced mathematical concepts.

In conclusion, of sin, cos, or tan, tan is the graph that is not differentiable at all points due to the vertical asymptotes in its graph.

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