What does \frac{dy}{dx} represent geometrically?

The gradient of the tangent to the curve at that point.

Write the first principles definition of f'(x).

f'(x)=\lim{h\to0}\frac{f(x+h)-f(x)}{h}

What is a chord of a curve?

A straight line joining two points on the curve.

Why is the chord gradient only an approximation to the tangent gradient?

It is an average gradient between two points; it equals the tangent gradient only in the limit as h\to0.

What does \frac{dy}{dx} measure in a modelling context?

The rate of change of y with respect to x.

What does \frac{dy}{dx}<0 tell you?

y is decreasing as x increases.

What is the geometrical meaning of f'(a)=0?

The tangent at x=a is horizontal.

The derivative of f'(x); the rate of change of the gradient.

The derivative as a gradient and rate of changeEdexcel International A Level Maths: Flashcards

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What does $\frac{\mathrm{d}y}{\mathrm{d}x}$ represent geometrically?

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What does $\frac{\mathrm{d}y}{\mathrm{d}x}$ represent geometrically?: The gradient of the tangent to the curve at that point.
Write the first principles definition of $f'(x)$ .: $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
What is a chord of a curve?: A straight line joining two points on the curve.
Gradient of the chord joining $x$ and $x+h$ ?: $\frac{f(x+h)-f(x)}{h}$
Why is the chord gradient only an approximation to the tangent gradient?: It is an average gradient between two points; it equals the tangent gradient only in the limit as $h\to0$ .
What does $\frac{\mathrm{d}y}{\mathrm{d}x}$ measure in a modelling context?: The rate of change of $y$ with respect to $x$ .
Units of $\frac{\mathrm{d}s}{\mathrm{d}t}$ if $s$ is in metres and $t$ in seconds?: m s $^{-1}$
What does $\frac{\mathrm{d}y}{\mathrm{d}x}<0$ tell you?: $y$ is decreasing as $x$ increases.
What is the geometrical meaning of $f'(a)=0$ ?: The tangent at $x=a$ is horizontal.
What is $f''(x)$ ?: The derivative of $f'(x)$ ; the rate of change of the gradient.
Alternative notation for the second derivative of $y$ ?: $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$
If $f''(x)>0$ , what is the gradient doing?: Increasing as $x$ increases.
First principles for $f(x)=x^2$ : simplified chord gradient?: $2x+h$ , so $f'(x)=2x$ .

Exam questions on The derivative as a gradient and rate of change

A curve has equation $y=x^2+3x$ . The point $P$ has coordinates $(2,10)$ and the point $Q$ on the curve has $x$ -coordinate $2+h$ , where $h\neq0$ .
Explain why the gradient of the chord $PQ$ is not equal to the gradient of the tangent at $P$ , and how the gradient of the tangent can be found from it.2 marks
The volume $V$ cm $^3$ of water in a tank at time $t$ minutes is modelled by $V=40+6t-0.5t^2$ for $0\le t\le10$ .
Find the time at which the volume of water momentarily stops changing, and explain what the sign of $\frac{\mathrm{d}V}{\mathrm{d}t}$ tells you about the volume just before this time.2 marks
The curve $y=f(x)$ has equation $f(x)=x^3-6x^2+9x+2$ .
Find $f'(x)$ and hence solve $f'(x)=0$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).