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Drag the red point in the display below to move the tangent line along the quartic (fourth order) curve.

You can see that the gradient function f'(x) of a quartic curve is a cubic curve.

Finally you can investigate higher order curves. In each case, the gradient function f'(x) will be a curve of one order lower.
 Select polynomial order:

## Summary/Background

A graphic calculator may not be allowed in your exam. However you can use one for revision. The TI-83, or similar model, can display graphs and their gradient function for you.
 To set this up, go to the Y= screen. Against Y1= enter a simple curve, 3x^2+1 for example. Against Y2= you should make the following selections: MATH 8 VARS Y-VARS 1 1 then type in ,X,X) The result should be that the Y= screen shows Y1=3x^2+1 and Y2=nDerive(Y1,X,X). Go to the GRAPH screen, to see the graphs of y=3x^2+1 and the gradient function. In this case the gradient function displayed will be that of y=6x. The calculator cannot tell you this though. You will have to work that out for yourself.If now you change the equation against Y1, the calculator will automatically show its derivative.
• The function that gives the gradient at any point on a given curve is called the gradient function for that curve
• If the curve is given by y=f(x), then the gradient function is often denoted by f'(x).
• If f(x) is quadratic, then f'(x) is linear
• If f(x) is cubic, then f'(x) is quadratic
• If f(x) is quartic (fourth degree), then f'(x) is cubic

JSXGraph is a cross-browser library for interactive geometry, function plotting, charting, and data visualization in a web browser. It is implemented completely in JavaScript, does not rely on any other library. It uses SVG and VML and is fully HTML5 compliant.

## Glossary

### cubic

A mathematical expression of the 3rd degree; having a x³ term but no higher one.

### derivative

rate of change, dy/dx, f'(x), , Dx.

### equation

A statement that two mathematical expressions are equal.

### function

A rule that connects one value in one set with one and only one value in another set.

The slope of a line; the angle of its inclination to the horizontal.

### graph

A diagram showing a relationship between two variables.
The diagram shows a vertical y axis and a horizontal x axis.

### linear

Straight, not curved. A linear equation is of the first degree, for example y = 2x+1.

### polynomial

an expression which has whole-numbered powers of the variable, x, multiplied by coefficients, and added together.
Examples: y = 2x, y = 1+x², y = 3-2x+7x³

A mathematical expression of the 2nd degree; having a x² term but no higher one.

### tangent

1. The trigonometrical function defined as opposite/adjacent in a right-angled triangle.
2. A straight line that touches a curve at one point.

### work

Equal to F x s, where F is the force in Newtons and s is the distance travelled and is measured in Joules.

Full Glossary List

## This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AP Calculus AB (USA)3DifferentiationSlope-
AP Calculus BC (USA)3DifferentiationSlope-
AQA A-Level (UK - Pre-2017)C1DifferentiationSlope-
CBSE XI (India)CalculusLimits and DerivativesDefinition of a derivative-
CCEA A-Level (NI)C1DifferentiationSlope-
CIE A-Level (UK)P1DifferentiationSlope-
Edexcel A-Level (UK - Pre-2017)C1DifferentiationSlope-