# Differential calculus: derivatives and power functions

Learning Material  |  Interactive Lesson

Published on 2018 August 13th

Description
Observe the non-linear distance-time graph of a rocket travelling at a changing velocity. Calculate the average velocity of the rocket over time intervals that become progressively shorter. Tabulate the results and derive a formula for finding the instantaneous velocity at a given point. In the second activity, observe the non-linear graphs of various powers of x and select the expressions for finding the gradient of the secant between small intervals of x (represented by ?x). Plot the derivatives of each function and look for patterns. This learning object is a combination of two objects in the same series.
Objective
Students apply the concept of a limit in the context of the rate of change of a non-linear function.
Students interpret the gradient of the tangent to a curve as being the rate of change of the function.
Students apply first principles methods to differentiate the functions of t² and various powers of x.
Students interpret the concept of a derivative of a function, and identify the derivative for t² and for various power functions of x.
Students investigate the derivative to calculate the range of a function at a nominated point.

#### Curriculum Information

 Education Type K to 12 Grade Level Grade 9, Grade 10 Learning Area Mathematics Content/Topic Patterns and Algebra Patterns and algebra Intended Users Learners, Students Competencies