Maths: Applications and Interpretation · Calculus. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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The gradient of a curve changes from point to point. To find it at one point, take a chord from that point to a nearby point and move the second point closer and closer. The value the chord gradient approaches is called a limit, and it is the gradient of the tangent there.
The derivative, written dy/dx or f′(x), is the function that gives this gradient for every x. It is also a rate of change: how fast y changes as x changes, in units of y per unit of x.
The sign of f′(x) tells you the direction of the graph: positive means going up from left to right, negative means going down.
Students confuse f(x) with f′(x). "f is increasing" is about the sign of f′(x), not about f(x) being positive. A function can be negative and increasing at the same time.
Answer: f is decreasing for −3 < x < 3.
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