Derivative Notation

A gradient that is a function

The gradient of $y=x^2$y = x² is 2 at x = 1, 4 at x = 2 and 6 at x = 3; at x = −2 it is −4, where the curve falls. At every x the gradient is 2x. So the gradient is not one number but a new function of x, made from the old one.

That new function is the derivative. It is used so often that it needs a short name, and two are in use: one from Lagrange and one from Leibniz. A third symbol, $\frac{d}{dx}$d/dx, is the instruction to find it.

The curve $y=x^2$y = x² and the line y = 2x, its gradient function. Above x = 3 the curve has gradient 6, and the line has height 6. Above x = −2 the curve falls with gradient −4, and the line has height −4.

Lagrange: f'(x)

Name the function f, so $f\left(x\right)=x^2$f(x) = x². Its derivative is written f'(x), read "f prime of x". The small mark after the f is the prime, and it means the derivative of f. So f'(x) = 2x.

Put a number in and f' gives the gradient at that point: f'(3) = 2 × 3 = 6 and f'(−1) = −2. Compare f(3) = 9, the height of the curve at x = 3. The prime is the whole difference between a height and a gradient.

Each function keeps its own letter. If a second function is named g, its derivative is g'(x), and g'(4) is its gradient at x = 4.

Leibniz: $\frac{dy}{dx}$dy/dx

Leibniz worked with y and x. A change in x is written $\Delta x$Δx, read "delta x", and the change in y that goes with it is $\Delta y$Δy. The gradient of a chord is $\Delta y/\Delta x$Δy / Δx.

On $y=x^2$y = x², from (1, 1) to (2, 4), $\Delta x=1$Δx = 1 and $\Delta y=3$Δy = 3, so $\Delta y/\Delta x=3$Δy / Δx = 3. Shrink the steps and the chord turns toward the tangent at (1, 1). The limit of $\Delta y/\Delta x$Δy / Δx as $\Delta x$Δx tends to 0 is written $\frac{dy}{dx}$dy/dx, read "d y by d x". For $y=x^2$y = x², $\frac{dy}{dx}=2x$dy/dx = 2x, so at (1, 1) it is 2.

$\frac{dy}{dx}$dy/dx is written like a fraction because it is the limit of one: a small rise over a small run. It is one symbol for that limit, not a division of a number called dy by a number called dx. $\frac{dy}{dx}$dy/dx and f'(x) name the same function, 2x, in two styles.

On $y=x^2$y = x², the chord from (1, 1) to (2, 4) has $\Delta x=1$Δx = 1 and $\Delta y=3$Δy = 3, so $\Delta y/\Delta x=3$Δy / Δx = 3. The less steep line is the tangent at (1, 1), where $\frac{dy}{dx}=2$dy/dx = 2.

$\frac{d}{dx}$d/dx: an instruction

Written on its own in front of an expression, $\frac{d}{dx}$d/dx is an instruction: differentiate what follows, with respect to x. So $\frac{d}{dx}\left(x^2\right)=2x$d/dx (x²) = 2x is read "the derivative of $x^2$x² with respect to x is 2x". When $y=x^2$y = x², $\frac{d}{dx}\left(y\right)$d/dx (y) is $\frac{dy}{dx}$dy/dx.

"With respect to x" names the variable that is changing. The letters can be others. If the area of a square of side r is $A=r^2$A = r², then $\frac{dA}{dr}=2r$dA/dr = 2r, the rate the area grows per unit of side; and if $s=t^2$s = t², then $\frac{ds}{dt}=2t$ds/dt = 2t, with $\frac{d}{dt}$d/dt as the instruction. The working is the same as for $x^2$x² every time.

$\frac{d}{dx}$d/dx is not a number and nothing is multiplied by it. $\frac{d}{dx}\left(x^2\right)$d/dx (x²) is not $x^2/x$x² / x or $x^2\times d$x² × d; it is the derivative of $x^2$x², which is 2x.

Differentiating twice

The derivative f'(x) = 2x is itself a function, so it has a derivative too. The graph of y = 2x is a straight line climbing 2 for every 1 across, so its gradient is 2 at every x. That is the second derivative of f, written f''(x), read "f double prime of x": f''(x) = 2.

In Leibniz's style the second derivative is $\frac{d^{2}y}{d{x}^2}$d²y/dx², read "d two y by d x squared". It is $\frac{d}{dx}$d/dx applied to $\frac{dy}{dx}$dy/dx, and the two 2s count how many times $\frac{d}{dx}$d/dx has been applied: the 2 sits on the d at the top and on the dx at the bottom. For $y=x^2$y = x², $\frac{d^{2}y}{d{x}^2}=2$d²y/dx² = 2.

Each prime and each 2 counts one more differentiation. f''(x) does not mean 2 × f'(x), which is 4x, and $\frac{d^{2}y}{d{x}^2}$d²y/dx² does not mean ${\left(\frac{dy}{dx}\right)}^2$(dy/dx)², which is $4x^2$4x².

The first derivative of $y=x^2$y = x² is the line y = 2x, which climbs 2 for every 1 across. Its gradient is therefore 2 everywhere, and the level line y = 2 is the second derivative. The two cross at (1, 2).

The usual mistakes

Reading f(x) as the derivative. Without the prime nothing has been differentiated: f(3) = 9 is the height and f'(3) = 6 is the gradient. A capital F names a different function altogether.

Reading $\frac{dy}{dx}$dy/dx as $y/x$y / x. For $y=x^2$y = x² at x = 3, $y/x=9/3=3$y / x = 9 / 3 = 3, but $\frac{dy}{dx}=6$dy/dx = 6. The gradient divides the small changes, not the values.

Turning $\frac{dy}{dx}$dy/dx upside down. $\frac{dx}{dy}$dx/dy is a small run over a small rise, which is 1 ÷ gradient.

Writing $\frac{d{y}^2}{d{x}^2}$dy²/dx² or ${\left(\frac{dy}{dx}\right)}^2$(dy/dx)² for the second derivative. At x = 3, ${\left(\frac{dy}{dx}\right)}^2=6^2=36$(dy/dx)² = 6² = 36, but $\frac{d^{2}y}{d{x}^2}=2$d²y/dx² = 2.

Reading f''(x) as f'(x) doubled or squared. Two primes mean differentiate twice: f''(x) = 2, while 2f'(x) = 4x.