- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Exponential and trigonometric functions practice question 7
Let . Find , the 50th derivative of .
Differentiate four times in a row. What happens after the fourth?
The derivatives repeat every four steps, so only the remainder when 50 is divided by 4 matters.
A formula in x. Use ^ for powers and brackets for every input: sin(2x), e^(x^2), log(x+1). Give exact values: 1/6, not 0.167.
Theory UBC Math 100 §4.2CLP-1 §2.8Key results
Worked solution
Strategy
The derivatives of sin x cycle with period 4. Find where the 50th falls in the cycle.
Working
After four derivatives, sin x returns.
Twelve full cycles bring us back to sin x at the 48th derivative.
Two more steps from sin x.
Answer
The answer is the 50th derivative of sin x is negative sin x.
Check
Each derivative shifts sine left by π/2, so the 50th is sin(x + 25π), which is −sin x.
Where this goes wrong
Answering sin x because 50 is even. The cycle has length 4, not 2: the 48th derivative is sin x, and two more steps give −sin x.
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