Pure: Differentiation
Differentiation is the process of finding the gradient function (derivative) of a curve, giving the rate of change or gradient at any point on it. A Level Maths covers differentiating polynomials, trig, exponential and log functions using the chain, product and quotient rules, then applying derivatives to find tangents, normals, stationary points and rates of change.
Before you start
Make sure you're comfortable with these topics first:
Method
- 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.
- 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 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.
- 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.
- 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.
- 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.
Worked example
The curve C has equation y = (4x - 1)^3. Find dy/dx, and hence find the gradient of C at the point where x = 1.
- Use the chain rule: differentiate the outer power first, treating (4x-1) as a single block: dy/dx = 3(4x-1)^2 x (derivative of 4x-1).
- The derivative of 4x - 1 is 4.
- Combine: dy/dx = 3(4x-1)^2 x 4 = 12(4x-1)^2.
- Substitute x = 1: dy/dx = 12(4(1)-1)^2 = 12(3)^2 = 12 x 9.
- Final answer: gradient = 108.
Practice questions
Try each question, then tap to reveal the answer.
Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
Find dy/dx for y = 4x^3 - 3/x^2 + 2sqrt(x), for x > 0.
The curve C has equation y = (2x - 3)^5. Find the equation of the tangent to C at the point where x = 2, giving your answer in the form y = mx + c.
The curve C has equation y = x^3 - 6x^2 + 9x + 2. Find the coordinates of the stationary points of C, and use the second derivative to determine the nature of each one.
Free printable worksheet
Want more practice on paper? Download the pure: differentiation worksheet pack - 9 pages of exam-style questions with a full mark scheme. No sign-up, no email wall - just the PDF, free for personal and classroom use.
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