5 · IB Maths AA
Chain and product rules
Pick a composite or product function, drag the point along the curve and watch the tangent gradient being assembled step by step from the chain rule or the product rule.
Building dy/dx at x =
Key equations
Tangent at x = a: y − f(a) = f′(a)(x − a)
Try this
- With y = sin(x²), move the point to x = 0. Why is the gradient zero even though cos(0) = 1? Look at du/dx in step 1.
- For y = x e⁻ˣ, find the x-value where the tangent is horizontal and check it by solving (1 − x)e⁻ˣ = 0.
- Turn on f′(x) for y = eˣ cos x and note where the green curve crosses zero; match each crossing to a turning point of the main curve.
Common error: differentiating only the outer function and forgetting to multiply by du/dx, or writing the derivative of uv as (du/dx)(dv/dx); write u, v and their derivatives down first, exactly as in steps 1 and 2.