Differentiation

Question 1

Equations of Differentiation

Answer 1

Leibniz:
y=ax^n
dy/dx=n x ax^n-1

Newton:
f(x)=ax^n
f ’(x)= n x ax^n-1

Question 2

How to find the rate of change?

Answer 2

1. Differentiate
2. Input the value
3. Solve

Question 3

How to find the rate of change?

Answer 3

1. Differentiate
2. Input the value
3. Solve

Question 4

How to find the equation of the tangent to a curve?

Answer 4

1. Find the coordinate point
2. Differentiate and find the gradient
3. Use the coordinate point and the gradient to find the equation of the line

Question 5

How to tell is a function is strictly increasing or decreasing at a certain point?

Answer 5

A function is strictly increasing if f ‘(x) > 0

A function is strictly decreasing if f‘(x) < 0

Question 6

What are the four types of curves you can find stationary points for?

Answer 6

1. Maximum turning point
2. Minimum turning point
3. Rising point of inflection
4. Falling point of inflection

Question 7

How to find stationary points?

Answer 7

1. Differentiate and find the roots
2. Input the roots into original equation and find the y coordinates.
3. Use a nature table and determine the slopes by inputting values just left and right of your roots into the differentiated equation.

Question 8

How to sketch the graph of a function?

Answer 8

1. Find the roots of the function.
2. Find the y intercept (when x=0)
3. Find the stationary points.
4. Find the y coordinates
5. Use nature tables.
6. Sketch the graph.

Question 9

What is a closed interval, and how to determine the maximum and minimum values of a function?

Answer 9

A closer interval is a defined interval on the domain (x-values) of the function.

To determine the maximum and minimum values of the function within a closed interval you must check:
1. The value of the function at any stationary points that lie within the interval (the y value)
2. The value of the function at the end points of the interval (input the values of closed interval as x into the original function)