- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Exponential and trigonometric functions practice question 6
Let for . Find .
Both cot and csc are built from sin and cos. How could you rewrite each one so that a rule you already know applies?
Write and , then use the quotient rule on each.
The derivatives of the co-functions cot and csc mirror those of tan and sec, with a minus sign.
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
Rewrite cot and csc in terms of sin and cos, differentiate with the quotient rule, and turn each result back into cot and csc.
Working
Quotient rule; the Pythagorean identity makes the numerator −1.
Quotient rule with a constant numerator.
Add the two derivatives.
Answer
The answer is f prime of x equals negative csc squared x minus csc x cot x.
Check
At x = π/2 the formula gives −1 − 0 = −1, and f(1.58) − f(1.56) ≈ −0.020 over a step of 0.02.
Where this goes wrong
Dropping both minus signs, as if cot and csc behaved like tan and sec. Both co-function derivatives are negative: (cot x)′ = −csc² x and (csc x)′ = −csc x cot x.
Leans on
The quotient rule, from Differentiation rules
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