Single Variable Calculus

Start Free Trial

Single Variable Calculus, Chapter 7, 7.2-2, Section 7.2-2, Problem 44

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Differentiate $f(x) = \ln (x^2 + x + 1)$. Check your answer if its reasonable by comparing the graphs of $f$ and $f'$.

$ \begin{equation} \begin{aligned} \text{if } f(x) =& \ln (x^2 + x + 1), \text{ then} \\ \\ f'(x) =& \frac{\displaystyle \frac{d}{dx} (x^2 + x + 1) }{x^2 + x + 1} \\ \\ f'(x) =& \frac{2x + 1}{x^2 + x + 1} \end{aligned} \end{equation} $

We see from the graph that $f$ is increasing when $f'(x)$ is positive. On the other hand, $f$ is decreasing whenever $f'(x)$ is negative. We can say that our answer is reasonable.

Posted on

Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial