The derivative: An intuitive introduction
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Derivative & the direction of a function
A function f of x is plotted below. Highlight an interval where f prime of x with the first derivative of f with respect to x is greater than 0. So if our derivative, f prime of x, is greater than 0, that means that the slope of the tangent line is positive. That means that the function must be increasing whenever this is true. So where is the function increasing? So over here we see that as x increases, our function is going to smaller and smaller values. So our function is decreasing this entire place. Then it's decreasing at slower and slower rates all the way to the point right over here where the slope of the tangent line is flat. And then the function starts increasing. And it starts increasing at faster and faster and faster rates. So we could put this blue area anywhere from here to anywhere in this region here where the function itself is increasing. Notice, the function could even be negative, but the function is increasing. So even this would be a valid place. That would be a valid place. Anywhere where the function's rate of-- when x increases, the function is actually getting larger. So let's check our answer. And let's do a couple more of these. A function f of x is plotted below. Highlight an interval where f of x is greater than 0. So f of x should be sitting in the first or second quadrant right over here. It needs to be greater than 0, and f prime of x is less than 0. So f prime of x less than 0 means that the function is decreasing. The slope of the tangent line is negative. So let's think about it. The two areas where f of x is greater than 0 are this interval right over here and this interval right over here. But we also care about the function decreasing. This won't be valid, because the function is increasing here. It's increasing at slower and slower rates, but it is increasing. The function is going up as x goes up. If we go over here, the function is decreasing as x increases. So this seems to meet our constraints. This area right over here won't because the function is positive, but it's also increasing. So here the derivative is positive. Here the derivative is positive. Here the derivative is negative. So the function itself is positive, but the slope of the tangent line is negative. It's decreasing as x increases. Let's do one more. The function f is plotted below. Highlight an interval where f of x is greater than 0 and f prime of x is less than 0. So the same exact idea-- where are we positive but our function is actually decreasing? So it's positive here, but we can't pick that. It's positive here, but we can't pick that. The function is decreasing here, but the function isn't positive there. The derivative is negative, but the function isn't positive. So that wouldn't make it. So this is the only region that we can throw it in. And we got it right again.