MTH6131 Chap.4 The Derivative from First Principles
The Derivative from First Principles
Derivatives are about rates of change
The differentiation notes open with a single claim: derivatives are about rates of change. If x changes from x₀ to x₀ + Δx, the function changes by Δy = f(x₀ + Δx) − f(x₀), and the ratio Δy/Δx is the average rate of change over that interval. The average speed of a car between two towns is the standard example.
What we usually want is the speed at one instant, and the way to get it is to shrink the interval.
From average to instantaneous
As Δx tends to zero, the average rate becomes the derivative, provided the limit exists. Geometrically the secant lines through two points of the curve turn into the tangent line at one point.
The method for computing it is always the same: expand f(x + Δx), subtract f(x), divide by Δx, and only then take the limit. Setting Δx to zero too early gives zero over zero and says nothing.
Quotients need a common denominator at the subtraction step, and square roots need the conjugate.
Differentiable means continuous, but not the reverse
The lecture proves that a function differentiable at a point is continuous there, by multiplying the difference quotient by Δx and letting both factors take their limits. The converse fails, and the lecture examples are chosen to show how.
The absolute value function has a corner at zero, where the left and right slopes differ. The cube root has a vertical tangent at zero, where the slope grows without bound. The square root of x + 1 starts at an endpoint, where only a one-sided slope exists.
Two sketches for every function
Each lecture example asks for the function and its derivative to be sketched together.
Reading slope from shape in this way is what later makes curve sketching quick: rising means a positive derivative, a smooth turning point means a zero, a corner means a jump in the derivative.
What this chapter covers
- 01
Increments and the average rate of change
- 02
The difference quotient and its limit
- 03
The expand, subtract, divide, limit procedure
- 04
Common denominators and conjugates
- 05
Why differentiable implies continuous
- 06
Corners, vertical tangents and endpoints
- 07
Sketching a function beside its derivative
Worked example · free
First principles on a reciprocal
- 2Write the increment Δy over a common denominator.
- 1Divide by Δx and cancel.
- 2Take the limit as Δx tends to zero.
Key terms
- Increment
- The change Δx in a variable as it moves from one value to another, and the resulting change Δy in a function of it.
- Average Rate of Change
- The ratio of the change in a function to the change in its input over an interval, the slope of the secant line.
- Difference Quotient
- The expression (f(x + Δx) − f(x))/Δx whose limit as Δx tends to zero defines the derivative.
- Tangent Line
- The line through a point of a curve whose slope is the derivative there, the limiting position of the secant lines.
- Vertical Tangent
- A tangent line parallel to the vertical axis, where the derivative grows without bound and so does not exist.
The Derivative from First Principles FAQ
Why can I not just put Δx equal to zero in the difference quotient?
Because the result is zero over zero, which carries no information. The derivative is a limit precisely because the quotient is undefined at Δx = 0. The procedure simplifies the quotient for non-zero Δx until the division disappears, and only then takes the limit, which is now a simple substitution.
Is every continuous function differentiable?
No. Differentiability implies continuity, but not the other way round. The absolute value function is continuous at zero yet has a corner there, with left slope minus one and right slope one, so it has no derivative at zero. The cube root is continuous at zero but has a vertical tangent there.
What units does a derivative have?
The units of the output divided by the units of the input. If distance is in kilometres and time in hours, the derivative of distance with respect to time is in kilometres per hour. Checking units is a quick way to confirm that a rate has been set up the right way round.
Exam move
Differentiate three functions from first principles, a polynomial, a reciprocal and a square root, and keep the working as a template for each type of rewrite. Then, for each lecture example, sketch the function and its derivative side by side, marking where the derivative is zero or fails to exist.
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