Algebra

Differentiation

Move a tangent and discover the power rule.

Predict, test and explain

Step 1 of 3

Predict

At x = 1, change f(x) = x² to f(x) = x² + 4. What happens to the tangent gradient?

Further exploration: introductory polynomial calculus.

f(x)=x^{2}

Scroll the diagram sideways to see every label.

f'(x)=2x

Tangent gradient

f'(1)=2

Chord gradient

\frac{f(t+h)-f(t)}{h}=3

Red is the tangent at t. Green dashes are the chord through t and t+h. Shrinking h makes the chord gradient approach the derivative. The power rule multiplies by the power and reduces that power by one; a constant differentiates to zero.

Point and tangent
Quantity	Value
t	1
f(t)	1
Tangent gradient	2
Tangent equation	$y=1+2(x-1)$

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

Find the tangent gradient at the given input.

$y=x^3-2x,\qquad x=2$

Hint

Differentiate each term before substituting the input value.

Reveal answer

The derivative gives the tangent gradient; substituting two gives ten.

$\frac{dy}{dx}=3x^2-2,\qquad \left.\frac{dy}{dx}\right|_{x=2}=10$

Build an explanation

Hint 1

The derivative describes the gradient of the tangent at each point. It measures how the output changes locally as the input changes.

Hint 2

For a polynomial term axⁿ with a positive integer n, differentiate to get a × n × xⁿ⁻¹. A constant term has derivative zero.

Hint 3

Differentiate each term, then substitute the chosen x-value. For a tangent at (x₀, y₀) with gradient m, use y − y₀ = m(x − x₀).

Worked example

This example uses fixed values, separate from the diagram controls.

For y = x³ − 2x, find dy/dx and the tangent at x = 1.

Differentiate the terms: dy/dx = 3x² − 2.
At x = 1, the tangent gradient is 3 × 1² − 2 = 1.
The point on the original curve is (1, 1³ − 2 × 1) = (1, −1).
Use y + 1 = 1(x − 1), which simplifies to y = x − 2.

Answer: dy/dx = 3x² − 2; tangent y = x − 2.

Connect this idea

Watch out: The derivative gives gradient, not the curve's height. A gradient of zero identifies a stationary point but does not by itself prove a maximum or minimum.