Define f'(x) from first principles.

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

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

The gradient of the tangent to the curve at a general point.

A straight line joining two points on a curve.

What happens to the gradient of chord PQ as Q\to P?

It tends to the gradient of the tangent at P.

What does \frac{dh}{dt} represent when h is height and t is time?

The instantaneous velocity (rate of change of height).

What is the second derivative?

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

Where is f'(x)=0 on the graph of y=f(x)?

At a minimum, maximum or other stationary point.

Gradient of a curve and differentiation from first principlesAQA A-Level Maths: Flashcards

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Define $\mathrm{f}'(x)$ from first principles.

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Define $\mathrm{f}'(x)$ from first principles.: $\mathrm{f}'(x)=\lim_{h\to0}\frac{\mathrm{f}(x+h)-\mathrm{f}(x)}{h}$
What does $\frac{\mathrm{d}y}{\mathrm{d}x}$ represent geometrically?: The gradient of the tangent to the curve at a general point.
What is a chord?: A straight line joining two points on a curve.
What happens to the gradient of chord $PQ$ as $Q\to P$ ?: It tends to the gradient of the tangent at $P$ .
Chord gradient for $y=x^2$ between $x$ and $x+h$ ?: $2x+h$
First principles result for $x^2$ ?: $2x$
First principles result for $x^3$ ?: $3x^2$
Expand $(x+h)^3$ .: $x^3+3x^2h+3xh^2+h^3$
Equation of tangent at $x=a$ ?: $y-\mathrm{f}(a)=\mathrm{f}'(a)(x-a)$
What does $\frac{\mathrm{d}h}{\mathrm{d}t}$ represent when $h$ is height and $t$ is time?: The instantaneous velocity (rate of change of height).
What is the second derivative?: $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=\mathrm{f}''(x)$ , the rate of change of the gradient.
If $y=x^3$ , what is $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$ ?: $6x$
Where is $\mathrm{f}'(x)=0$ on the graph of $y=\mathrm{f}(x)$ ?: At a minimum, maximum or other stationary point.

Exam questions on Gradient of a curve and differentiation from first principles

The curve $C$ has equation $y=x^2$ . The point $P(3,9)$ lies on $C$ , and $Q$ is the point on $C$ with $x$ -coordinate $3+h$ , where $h\neq0$ .
Find the equation of the tangent to $C$ at $P$ .2 marks
The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=x^3$ .
Find the coordinates of the points on the curve $y=\mathrm{f}(x)$ at which the gradient is 12.2 marks
The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=3x^2-5x$ .
Prove from first principles that $\mathrm{f}'(x)=6x-5$ .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).