- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Exponential and trigonometric functions practice question 13
Each working below writes and differentiates it. Which one is correct at every step?
Two of the workings end in sec² x, so where a working ends proves nothing. Check every step of each against the quotient rule, sign by sign.
With and : and . The quotient rule is .
Theory UBC Math 100 §4.2CLP-1 §2.8Key results
Worked solution
Strategy
Apply the quotient rule to sin x over cos x with the derivative of cos x carrying its minus sign, then use cos² x + sin² x = 1. Judge each working by its steps, not by where it ends.
Working
Name the parts. The derivative of cosine is minus sine.
u′v − uv′ over v². The two minus signs make a plus.
The Pythagorean identity makes the numerator 1.
Answer
The answer is the derivative of tan x is sec squared x.
Check
tan x is increasing, so its slope at π/4 must be positive: sec² x gives 2 there, but −sec² x gives −2, −cot x gives −1 and (cos²x − sin²x)/cos²x gives 0.
Where this goes wrong
Picking a working because it ends in sec² x. One of them gets there with the derivative of cos x taken as +sin x, which leaves cos² x − sin² x on top; the minus sign is what produces cos² x + sin² x = 1.
Leans on
The quotient rule, from Differentiation rules
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