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.
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 note about graphic calculators
A graphic calculator may not be allowed in your exam. However you can use one for revision. The TI83, or similar model, can display graphs and their gradient function for you.
A graphic calculator may not be allowed in your exam. However you can use one for revision. The TI83, 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:  

 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
Software/Applets used on this page
This page uses JSXGraph.
JSXGraph is a crossbrowser 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.
This page also uses the MathJax system for displaying maths symbols.
JSXGraph is a crossbrowser 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.
This page also uses the MathJax system for displaying maths symbols.
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.
gradient
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.
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 wholenumbered powers of the variable, x, multiplied by coefficients, and added together.
Examples: y = 2x, y = 1+x²,y = 32x+7x³
Examples: y = 2x, y = 1+x²,
quadratic
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 rightangled triangle.
2. A straight line that touches a curve at one point.
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.
This question appears in the following syllabi:
Syllabus  Module  Section  Topic 

AP Calculus AB (USA)  3  Differentiation  Slope 
AP Calculus BC (USA)  3  Differentiation  Slope 
AQA ALevel (UK  Pre2017)  C1  Differentiation  Slope 
AQA AS Maths 2017  Pure Maths  Differentiation  Gradients 
AQA AS/A2 Maths 2017  Pure Maths  Differentiation  Gradients 
CBSE XI (India)  Calculus  Limits and Derivatives  Definition of a derivative 
CCEA ALevel (NI)  C1  Differentiation  Slope 
CIE ALevel (UK)  P1  Differentiation  Slope 
G Differentiation  G1 Differentiation for Gradients  Gradients  
Edexcel ALevel (UK  Pre2017)  C1  Differentiation  Slope 
Edexcel AS Maths 2017  Pure Maths  Differentiation  Gradients 
Edexcel AS/A2 Maths 2017  Pure Maths  Differentiation  Gradients 
I.B. Higher Level  6  Differentiation  Slope 
I.B. Standard Level  6  Differentiation  Slope 
Methods (UK)  M8  Differentiation  Slope 
I.B. (MSSL)  7  Differentiation  Slope 
OCR ALevel (UK  Pre2017)  C1  Differentiation  Slope 
OCR AS Maths 2017  Pure Maths  Differentiation Basics  Gradients 
OCR MEI AS Maths 2017  Pure Maths  Differentiation Basics  Gradients 
OCRMEI ALevel (UK  Pre2017)  C2  Differentiation  Slope 
PreU ALevel (UK)  4  Differentiation  Slope 
Scottish (Highers + Advanced)  HM1  Differentiation  Slope 
Scottish Highers  M1  Differentiation  Slope 
Universal (all site questions)  D  Differentiation  Slope 
WJEC ALevel (Wales)  C1  Differentiation  Slope 