NUST NET (Engineering) · Maths · Differentiation. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
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.
Try each one, then open the answer.
Answer: C. f′(x) = 6x2 − 18x + 12 = 6(x − 1)(x − 2). f″(x) = 12x − 18 is −6 at x = 1 (maximum) and +6 at x = 2 (minimum). At x = 3/2, f″ = 0, which is not a stationary point.
Answer: B. A tangent parallel to the x-axis has gradient 0: 2x − 4 = 0, so x = 2 and y = 4 − 8 + 5 = 1. The other points are on the curve but have non-zero gradient.
Answer: A. With y = √x, dy = dx/(2√x) = 0.1/(2 × 5) = 0.01, so √25.1 ≈ 5 + 0.01 = 5.01. Forgetting the 2 in 2√x gives 5.02.
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