Pure: Differentiation

Method

1. Differentiate each term of a polynomial using the rule axⁿ → anx^n-1, rewriting roots and fractions as powers of x first if needed.Differentiate each term of a polynomial using the rule ax^n -> anx^(n-1), rewriting roots and fractions as powers of x first if needed.
2. For a composite function such as (expression)ⁿ, sin(expression) or e^expression, use the chain rule: differentiate the outer function, then multiply by the derivative of the inner expression.For a composite function such as (expression)^n, sin(expression) or e^(expression), use the chain rule: differentiate the outer function, then multiply by the derivative of the inner expression.
3. For a product of two functions u and v, use the product rule: dy/dx = u'v + uv'. For a quotient u/v, use the quotient rule: dy/dx = (u'v - uv')/v².For a product of two functions u and v, use the product rule: dy/dx = u'v + uv'. For a quotient u/v, use the quotient rule: dy/dx = (u'v - uv')/v^2.
4. Learn the standard derivatives: sin(x) → cos(x), cos(x) → -sin(x), e^x → e^x, and ln(x) → 1/x, applying the chain rule whenever the argument is not simply x.Learn the standard derivatives: sin(x) -> cos(x), cos(x) -> -sin(x), e^x -> e^x, and ln(x) -> 1/x, applying the chain rule whenever the argument is not simply x.
5. To find a tangent or normal at a point, substitute the x-value into the derivative to get the gradient, then use y - y1 = m(x - x1) with the tangent gradient m, or the negative reciprocal of m for the normal.
6. To find and classify stationary points, set dy/dx = 0 and solve for x, then use the second derivative (positive means minimum, negative means maximum) or check the sign of the gradient either side.

Practice questions

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