Calculate the derivative of the logarithmic function h(x) = ln(x^2 + 5x).
Answer and explanation
Correct answer: C. (2x + 5) / (x^2 + 5x)
- Step 1: Apply the logarithmic differentiation rule: d/dx [ln(u)] = u' / u.
- Step 2: Identify the argument: u = x^2 + 5x.
- Step 3: Differentiate the argument: u' = 2x + 5.
- Step 4: Combine into the fraction: h'(x) = (2x + 5) / (x^2 + 5x).
Answer
(2x + 5) / (x^2 + 5x).
Why not the others: '1 / (x^2 + 5x)' neglects the chain rule derivative of the argument; '(2x) / (x^2 + 5x)' differentiates only the leading quadratic term; '(x + 5) / (x^2 + 5x)' drops the factor of 2 from the derivative of x^2.
Practise more Calculus and Mathematical Analysis.
Free questions with an explanation for every one, and full timed General Education Diploma (GED Oman) mock tests.
More Calculus and Mathematical Analysis questions
- Evaluate the trigonometric limit: lim (x -> 0) [sin(7x) / (2x)].
- Find the derivative of the rational function f(x) = (3x - 5) / (2x + 1) for x != -1/2.
- What is the slope of the tangent line to the curve y = 2x^3 - 5x^2 + 4 at x = 2?
- Determine the indefinite integral: integral of (6x^2 - 4x + 5) dx.
- Evaluate the definite integral: integral from 0 to 3 of (4x + 1) dx.
- Find the local maximum of the polynomial function f(x) = -x^3 + 3x + 2.