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The figure shows the graph of π.
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What can be said about the differentiability of π at π₯ equals negative four?
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Here, weβve been given a graph which is defined over the interval from negative seven to negative one.
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Through this interval, we see that our curve is smooth at all points, aside from the point where π₯ is equal to negative four.
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At this point of coordinates negative four, five, we have a sharp corner.
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This means that the slope of the tangent, just to the left of π₯ equals negative four, will be different to the slope of the tangent just to the right of π₯ equals negative four.
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Here, we can even go so far as to say that one of our slopes will be positive and one of our slopes will be negative.
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Given that we have two different tangents on either side, it follows that it is not possible to define a tangent at π₯ equals negative four.
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And therefore, itβs also not possible to define the derivative.
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If we were to imagine the graph of π¦ equals π dash of π₯ our first derivative, we would expect to see a sharp jump in the π¦-value when π₯ equals negative four.
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From our observations, we conclude that the function is not differentiable at π₯ equals negative four because the functions rate of change is different on both sides of that point.
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And with this statement, we have answered our question.