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Sketching the derivative of a function is an illuminating task that helps us visualize the rate of change at every point along the function's graph. To begin sketching, we observe where the function is increasing or decreasing. An increasing function leads to a derivative that is positive, indicating that as you move along the function, your elevation is going up.

When sketching, it's essential to remember that the contour of the original function's graph suggests how steep or gentle the slopes will be. Gentle inclines translate to small positive values, while steep inclines correspond to larger positive values. Similarly, in declining sections, steepness reflects large negative values. By smoothly connecting these values, bearing in mind any maxima or minima as points where the graph crosses the x-axis, you'll create the rough graph of the derivative.

Local maxima and minima are the high and low points on a graph where the function changes direction. At these points, the derivative is zero because the slope of the tangent is horizontal. Identifying these points involves finding where the graph peaks (maxima) or troughs (minima).

To locate these on the derivative graph, we look for where the graph crosses the x-axis and changes signs. For instance, if the derivative changes from positive to negative, we've just passed a maximum. Conversely, when it changes from negative to positive, we've just encountered a minimum. It's crucial to mark these points clearly on the graph of the derivative, as they are significant in understanding the function's behavior.

Differentiability refers to whether a function has a derivative at every point within its domain. If a function is differentiable, it means that it has a tangent line at every point, and therefore, it can tell a coherent story of slopes across its graph. However, when a function has a sharp corner, a cusp, or a vertical tangent, it is not differentiable at that point because the slope is not consistent.

In the graph of the derivative, these nondifferentiable points appear as breaks or discontinuities. When sketching the graph of the derivative, keenly note these disruptions and do not draw a connecting line through them. This accurate portrayal of the derivative's continuity is a pivotal aspect of understanding the function's overall differentiability.

Tangent line slopes are the visual representation of a derivative at a specific point on a function. Imagine drawing a straight line that just skims a curve at a point. This line represents the instantaneous rate of change of the function at that point – in other words, its derivative.

By observing the slope of this tangent line, we can infer the derivative's value: positive slopes indicate a positive derivative, negative slopes indicate a negative derivative, and a flat (horizontal) line indicates a zero derivative. As you sketch the derivative, consider the steepness of these tangent lines as they will guide the rise and fall of your sketch across the derivative's graph. Mildly sloped tangents correspond to points close to the x-axis on the derivative graph, while steeper tangents lead to points that are further away from the x-axis, portraying the function's rapidly changing rate.