Differentiation and the tangent

Years 12 and 13

The derivative gives the gradient of a curve at any point, which is the gradient of its tangent.

The curve y = x² with a tangent at x = 1. The gradient there is 2.

For learners

A curve changes steepness as you move along it. At each point we can ask: how steep is it right here?

A tangent is a straight line that just touches the curve at one point and has the same steepness there.

Differentiation is a way of finding that steepness with a rule. For $y = x^{2}$ , the steepness at any point is $2 x$ . At $x = 1$ the gradient is 2.

A worked example

Find the gradient of $y = x^{2} + 3 x$ at $x = 2$ .

Differentiate: $\frac{d y}{d x} = 2 x + 3$ .
Put in $x = 2$ : $2 \times 2 + 3 = 7$ .
The gradient is 7.

For grown-ups

The derivative $\frac{d y}{d x}$ is the rate of change of $y$ with respect to $x$ . The rule for powers is: multiply by the power, then reduce the power by 1, so $x^{n}$ becomes $n x^{n - 1}$ .

Common mistakes

Forgetting to reduce the power.
Differentiating a constant and getting the constant back (it becomes 0).
Using $x$ in the answer when asked for the gradient at a particular point.

Why it matters

Differentiation describes speed and acceleration, the best shape for a box, and how fast things grow or cool.

Topics in quarter that use this

Differentiating polynomials
Differentiating with the chain rule
The product and quotient rules
Finding stationary points
Differentiating exponentials and trig
Kinematics using calculus