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 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
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
derivative
equation
function
gradient
graph
The diagram shows a vertical y axis and a horizontal x axis.
linear
polynomial
Examples: y = 2x, y = 1+xÂ²,
quadratic
tangent
2. A straight line that touches a curve at one point.
work
This question appears in the following syllabi:
Syllabus  Module  Section  Topic  Exam Year 

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   
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   