- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Differentiation rules practice question 6
Let . Find .
The denominator is a single power of , so divide it into each term of the numerator before differentiating.
Dividing gives . The power rule works for negative exponents too: .
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 §4.1CLP-1 §2.6Key results
Worked solution
Strategy
A quotient whose denominator is a single power of x splits into a sum of powers. That avoids the quotient rule entirely; the quotient rule gives the same answer with more algebra.
Working
Divide each term of the numerator by x².
Power rule on each term; each negative exponent brings down a negative factor.
Back to fractions, over a common denominator if you like.
Answer
The answer is f prime of x equals 1 plus 2 over x squared minus 10 over x cubed.
Check
For large x, f(x) ≈ x and the formula tends to 1; at x = 1 it gives −7, and (f(1.01) − f(0.99))/0.02 ≈ −7.00.
Where this goes wrong
Differentiating the top and the bottom separately, which gives (3x² − 2)/(2x). A quotient is not differentiated piece by piece: split it into powers first, or use the quotient rule.
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