State the first-principles definition of f'(x).

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

What does \frac{f(x+h)-f(x)}{h} represent?

The gradient of the chord between x and x+h.

What does \frac{dy}{dx} measure?

The rate of change of y with respect to x.

First principles result for f(x)=x^{3}?

f'(x)=3x^{2} (via 3x^{2}+3xh+h^{2}\to3x^{2}).

Why must you cancel h before taking the limit?

Otherwise you get \frac{0}{0}, which is undefined.

What is the second derivative?

f''(x) or \frac{d^{2}y}{dx^{2}}: the derivative of f'(x), the rate of change of the gradient.

Notation for velocity and acceleration from s?

v=\frac{ds}{dt}, a=\frac{d^{2}s}{dt^{2}}

What does f'(x) equal at a turning point?

0, so the gradient function crosses the x-axis there.

Differentiation from first principlesEdexcel A-Level Maths: Flashcards

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What is the gradient of a curve at a point?

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What is the gradient of a curve at a point?: The gradient of the tangent to the curve at that point.
State the first-principles definition of $f'(x)$ .: $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
What does $\frac{f(x+h)-f(x)}{h}$ represent?: The gradient of the chord between $x$ and $x+h$ .
What does $\frac{\mathrm{d}y}{\mathrm{d}x}$ measure?: The rate of change of $y$ with respect to $x$ .
First principles result for $f(x)=x^{2}$ ?: $f'(x)=2x$ (via $2x+h\to2x$ ).
First principles result for $f(x)=x^{3}$ ?: $f'(x)=3x^{2}$ (via $3x^{2}+3xh+h^{2}\to3x^{2}$ ).
Expand $(x+h)^{3}$ .: $x^{3}+3x^{2}h+3xh^{2}+h^{3}$
Why must you cancel $h$ before taking the limit?: Otherwise you get $\frac{0}{0}$ , which is undefined.
What is the second derivative?: $f''(x)$ or $\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}$ : the derivative of $f'(x)$ , the rate of change of the gradient.
If $f''(x)>0$ on an interval, what is happening to the gradient?: It is increasing.
Notation for velocity and acceleration from $s$ ?: $v=\frac{\mathrm{d}s}{\mathrm{d}t}$ , $a=\frac{\mathrm{d}^{2}s}{\mathrm{d}t^{2}}$
What does $f'(x)$ equal at a turning point?: $0$ , so the gradient function crosses the $x$ -axis there.
Where a curve is falling, what is the sign of $f'(x)$ ?: Negative.

Exam questions on Differentiation from first principles

The point $P(3,9)$ lies on the curve $y=x^{2}$ . The point $Q$ on the same curve has $x$ -coordinate $3+h$ , where $h\neq0$ .
Find the equation of the tangent to the curve at $P$ .2 marks
Let $f(x)=x^{3}$ .
Given that $f'(x)=3x^{2}$ , find $f''(x)$ , and hence find the rate of change of the gradient of the curve $y=f(x)$ when $x=2$ .2 marks
A curve has equation $y=x^{2}$ .
Prove, from first principles, that $\frac{\mathrm{d}y}{\mathrm{d}x}=2x$ .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).