- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Differentiation rules practice question 5
Let for . Find .
A square root, a cube root and a reciprocal: each is a power of x in disguise. What power is each one, before you differentiate anything?
Write every term as a power of first: and . The power rule holds for any constant exponent.
, since exponents add when powers of multiply.
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
Roots and reciprocals are powers in disguise. Rewrite each term as a constant times a power of x, then apply the power rule term by term.
Working
Add exponents for the product; a reciprocal makes the exponent negative.
Bring the exponent down and subtract one: 5/2 − 1 = 3/2 and −1/3 − 1 = −4/3.
Either form is fine; the second puts the roots back.
Answer
The answer is f prime of x equals five halves x to the three halves, plus 2x to the negative four thirds.
Check
At x = 1 the formula gives 4.5, and (f(1.01) − f(0.99))/0.02 ≈ 4.50; it is positive because both x²√x and −6/∛x increase.
Where this goes wrong
Differentiating x² and √x separately and multiplying, 2x · 1/(2√x) = √x. Combine the powers first, x²√x = x5/2, and differentiate once.
Leans on
Laws of exponents, from Functions, graphs and inverses
Try a similar question
Let . Find .Saved in this browser only.