Is differentiation from first principles required in DSE M1?

No. You need to recognise the definition of the derivative as a limit, but the curriculum notes say finding derivatives from first principles is not required in M1. (It is required in M2.)

What does the derivative represent?

The slope of the tangent to the curve at a point, which is also the instantaneous rate of change of y with respect to x.

Can a limit fail to exist?

Yes. For example 1/x has no limit as x tends to 0.

Limits and the Derivative of a Function (DSE M1)

The derivative of f(x) is f′(x) = lim_h→0 [f(x + h) − f(x)] / h, the slope of the tangent to the curve. In M1 you need the idea and the notation (y′, f′(x), dy/dx); finding derivatives from first principles is not required.

Solve Limits and the Derivative of a Function questions with AI

As Q slides towards P(1, −1) on y = x³ − 2x, the secant PQ turns into the tangent y = x − 2 of Example 1. Try it yourself →

Limits and Tangents: Key Facts

Definition of the derivativef′(x) = lim_h→0 [f(x + h) − f(x)] / h

Slope of the tangent at x = x₀f′(x₀)

Tangent liney − f(x₀) = f′(x₀)(x − x₀)

Rational function at infinityDivide the top and bottom by the highest power of x

0/0 formFactorise and cancel before substituting

Worked Examples — Limits and the Derivative of a Function

Example 1: Equation of a tangent

Find the equation of the tangent to the curve y = x³ − 2x at the point where x = 1.

dy/dx = 3x² − 2
At x = 1: slope = 3 − 2 = 1, and y = 1 − 2 = −1
y − (−1) = 1(x − 1)

Answer: y = x − 2

Example 2: Two standard limits

Find (a) lim_x→2 (x² − 4)/(x − 2) and (b) lim_x→∞ (3x² + 1)/(x² − 4).

(a) (x² − 4)/(x − 2) = x + 2 for x ≠ 2, so the limit is 4
(b) Divide by x²: (3 + 1/x²)/(1 − 4/x²)
As x → ∞ the fractions tend to 0, so the limit is 3

Answer: (a) 4; (b) 3

💡 Exam Traps & Secrets

⚠️ dy/dx is not a fractiond/dx is an operator meaning "differentiate with respect to x". The curriculum notes single this out: dy/dx is one symbol, not dy divided by dx.

⚠️ A tangent needs a point and a slopeSubstitute x₀ into dy/dx for the slope and into the original equation for the y-coordinate. Using the derivative for both is a common slip.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Tangent)Find the equation of the tangent to y = x² − 3x + 5 at x = 2 and draw it.

(0/0 limit)Find the limit of (x² − x − 6)/(x − 3) as x tends to 3.

(Limit at infinity)Find the limit of (2x + 5)/(x − 1) as x tends to infinity.