- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
The chain rule and logarithms practice question 5
Let . Find .
The square root covers the whole of . What is the outer function, what is the inside, and what must the outer derivative be multiplied by?
Write the root as a power, , with . The chain rule gives .
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 §4.3CLP-1 §2.9Key results
Worked solution
Strategy
A square root of a polynomial is a composition: the power ½ outside, 3x² + 4 inside. Differentiate the outside with the inside left alone, then multiply by the derivative of the inside.
Working
A square root is the power ½.
Power rule on the outside, then times the inner derivative 6x.
Half of 6x is 3x, and the power −½ puts the root in the denominator.
Answer
The answer is 3x over the square root of 3x² + 4.
Check
For large x, f(x) ≈ √3·x, a line of slope √3 ≈ 1.73, and 3x/√(3x² + 4) does tend to 3/√3 = √3.
Where this goes wrong
Writing 1/(2√(3x² + 4)) and stopping. That differentiates the root but not the 3x² + 4 inside it; the chain rule multiplies by its derivative, 6x.
Leans on
Fractional exponents, from Power functions and first sketches
Try a similar question
Let . Find at .Saved in this browser only.