9231 · 2.3
Differentiation flashcards
Revision flashcards for Cambridge 9231 Differentiation (syllabus 2.3). Flip, recall, then mark a real past-paper question.
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What is implicit differentiation?
A technique used to differentiate relations where the dependent variable (usually y) is not explicitly isolated on one side of the equation. We differentiate term-by-term with respect to x.
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When differentiating an expression in `y` with respect to `x`, what rule must you always apply?
The chain rule. For example, `d/dx(f(y)) = f'(y) * dy/dx`.
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What is `d/dx(y^n)`?
`n * y^(n-1) * dy/dx`
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What is `d/dx(xy)`?
You must use the product rule: `x * (dy/dx) + y * 1 = x(dy/dx) + y`.
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What is the formula for the first derivative of a parametric function `x=f(t), y=g(t)`?
`dy/dx = (dy/dt) / (dx/dt)`
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What is the formula for the second derivative of a parametric function?
`d²y/dx² = d/dx(dy/dx) = [d/dt(dy/dx)] / (dx/dt)`. Note that `dt/dx = 1 / (dx/dt)`.
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What is a common, incorrect formula used for the parametric second derivative?
Calculating `(d²y/dt²) / (d²x/dt²)`. This is completely wrong and will score no marks.
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How do you find the coordinates of a stationary point on a curve?
Set `dy/dx = 0` and solve for the variable(s). Then substitute back into the original equation(s) to find the corresponding coordinate(s).
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What does it signify if `dx/dt = 0` but `dy/dt ≠ 0` for a parametric curve?
The curve has a vertical tangent at that point, as the gradient `dy/dx` is undefined.
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How do you find the equation of a normal to a curve at a point `(x₁, y₁)`?
1. Find the gradient of the tangent, `m_T = dy/dx` at that point. 2. Find the gradient of the normal, `m_N = -1 / m_T`. 3. Use the formula `y - y₁ = m_N(x - x₁)`.