The definition of the derivative practice question 29
A student uses the definition of the derivative to find for the function below, as shown. Find the first line that is wrong.
Test each line twice: is the algebra right, and does the line match the rule its note names?
Write the definition of from memory, then compare it with line 1 symbol by symbol.
Which value is subtracted in the numerator: , or ?
Theory UBC Math 100 §3.3CLP-1 §2.2Key results
Corrected working
Strategy
Form f(2 + h) − f(2), expand, cancel the h, and let h → 0.
Working
f(2) = 7.
Expand; the constants cancel; divide every term by h.
Let h → 0.
Answer
The answer is f prime of 2 equals 4.
Check
f(2.01) − f(2) = 7.0401 − 7 = 0.0401 over a step of 0.01, a slope of 4.01, close to 4.
Where this goes wrong
Subtracting f(h) instead of f(2). The quotient compares the function at 2 + h with the function at 2; nothing else makes the h cancel.
The idea behind it
The definition of the derivative lesson, a figure to move, in a few steps
Leans on
Expanding a square, from Functions, graphs and inverses
Try a similar question
Use the limit definition of the derivative to find for .Saved in this browser only.