- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Differentiation rules practice question 1
Let . Find .
f is a sum of three terms: two powers of x with coefficients, and a lone 7. What happens to each kind of term when you differentiate?
Differentiate term by term with the power rule: , and a constant multiple stays in front.
The derivative of a constant is zero.
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 §2.4Key results
Worked solution
Strategy
The derivative of a sum is the sum of the derivatives, so take each term on its own with the power rule.
Working
Bring each exponent down as a factor and lower it by one.
A constant does not change, so its rate of change is zero.
Combine the pieces, keeping the minus sign.
Answer
The answer is f prime of x equals 12x cubed minus 10x.
Check
f is even, so f′ must be odd, as 12x³ − 10x is; at x = 1 it gives 2, close to (f(1.01) − f(0.99))/0.02 ≈ 2.001.
Where this goes wrong
Bringing each exponent down but keeping it, which gives 12x⁴ − 10x². The power rule also lowers every exponent by one: x⁴ becomes 4x³.
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