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 dy/dx is the rate at which y changes with x. In the test you never work from first principles: you recognise the shape of the function and apply the matching rule.
Ask three questions. Is it two functions multiplied (product rule)? Is it one divided by another (quotient rule)? Is it a function inside another function (chain rule)? For the chain rule, differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.
When x and y are mixed in one equation, differentiate every term with respect to x and attach dy/dx to each y term. When x and y are both given in terms of t, divide the two derivatives.
Forgetting the derivative of the inside in the chain rule, for example writing d/dx (sin 3x) = cos 3x instead of 3 cos 3x. Every time the inside is not just x, multiply by its derivative.
Answer: dy/dx = −5
Try each one, then open the answer.
Answer: D. dy/dθ = a cos θ and dx/dθ = −a sin θ, so dy/dx = −cos θ/sin θ = −cot θ. Missing the minus sign from the derivative of cos θ gives cot θ.
Answer: C. Take logs: ln y = x ln x, so (1/y) dy/dx = ln x + 1 and dy/dx = xx(1 + ln x). The power rule and the ax rule each give only part of this.
Answer: B. dy/dx = (dy/dt) ÷ (dx/dt) = 3t2/(2t) = 3t/2. Dividing the other way round gives 2/(3t).
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