NUST NET Maths · Maths · Differentiation. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The derivative f′(a) is the gradient of the tangent to the curve at x = a. The normal is the line at right angles to the tangent at the same point, so its gradient is −1/f′(a).
The sign of f′(x) tells you the behaviour of the function: positive means increasing, negative means decreasing. Where f′(x) = 0 the curve has a stationary point, and the second derivative tells you whether it is a maximum or a minimum.
A derivative is also a rate. With time as the variable, link two rates using the chain rule. For a small change δx, the change in y is close to f′(x) × δx.
Giving the x-value when the question asks for the maximum or minimum value. The value is f(x) at that point, so substitute x back into the original function, not into f′.
Answer: The local minimum value is 1, at x = 3.
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