Differentiation and gradients

Differentiation measures how quickly one quantity changes compared with another. On a graph, the derivative gives the gradient of the curve at a particular point.

From average to instantaneous gradient

Between two points, the average gradient is

$\frac{\text{change in }y}{\text{change in }x}=\frac{\Delta y}{\Delta x}.$

For a curve, this gradient changes as the points move. Bringing the two points arbitrarily close produces the gradient of the tangent: the instantaneous gradient.

The derivative of $y$ with respect to $x$ is written

$\frac{dy}{dx}.$

The power rule

For a constant power $n$,

$\frac{d}{dx}(x^n)=nx^{n-1}.$

Multiply by the old power, then reduce the power by one. For example,

$\frac{d}{dx}(x^5)=5x^4$

and

$\frac{d}{dx}(3x^4)=12x^3.$

The rule also works for negative and fractional indices wherever the function is differentiable:

$\frac{d}{dx}(x^{-2})=-2x^{-3},$

$\frac{d}{dx}(\sqrt{x})=\frac{d}{dx}(x^{1/2})=\frac12x^{-1/2}.$

Tangents and normals

For $y=x^2+3x$, the derivative is

$\frac{dy}{dx}=2x+3.$

At $x=2$, the tangent gradient is $2(2)+3=7$. The point on the curve is $(2,10)$, so the tangent is

$y-10=7(x-2).$

The normal is perpendicular to the tangent. Its gradient is the negative reciprocal, $-1/7$.

Common mistakes

• Differentiate each term separately; do not change addition into multiplication.
• A constant differentiates to zero.
• Find the point on the original curve, not on the derivative.
• Perpendicular gradients multiply to $-1$, so the normal gradient is a negative reciprocal.