Chapter 2: Q 22. (page 222)

Find the derivatives of the functions: $f (x) = \ln (\frac{x^{3}}{x^{2} + x + 1})$

Short Answer

Expert verified

The required answer is $\frac{x^{2} + 2 x + 3}{x (x^{2} + x + 1)}$ .

Step by step solution

Step 1. Given information.

The given function is: $f (x) = \ln (\frac{x^{3}}{x^{2} + x + 1})$

Step 2. Find the derivative of the given function.

$f (x) = \ln (\frac{x^{3}}{x^{2} + x + 1}) f' (x) = \frac{d}{d x} [\ln (\frac{x^{3}}{x^{2} + x + 1})] = \frac{1}{\frac{x^{3}}{x^{2} + x + 1}} \frac{d}{d x} (\frac{x^{3}}{x^{2} + x + 1}) = \frac{x^{2} + x + 1}{x^{3}} \frac{(x^{2} + x + 1) \frac{d}{d x} (x^{3}) - x^{3} \frac{d}{d x} (x^{2} + x + 1)}{{(x^{2} + x + 1)}^{2}} = \frac{x^{2} + x + 1}{x^{3}} \frac{3 x^{2} (x^{2} + x + 1) - x^{3} (2 x + 1)}{{(x^{2} + x + 1)}^{2}} = \frac{3 x^{4} + 3 x^{3} + 3 x^{2} - 2 x^{4} - x^{3}}{x^{3} (x^{2} + x + 1)} = \frac{x^{4} + 2 x^{3} + 3 x^{2}}{x^{3} (x^{2} + x + 1)} = \frac{x^{2} (x^{2} + 2 x + 3)}{x^{3} (x^{2} + x + 1)} = \frac{(x^{2} + 2 x + 3)}{x (x^{2} + x + 1)}$

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Find the derivatives of the functions: $f (x) = \ln (\frac{x^{3}}{x^{2} + x + 1})$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the derivative of the given function.

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Find the derivatives of the functions:f(x)=lnx3x2+x+1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the derivative of the given function.

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Find the derivatives of the functions: $f (x) = \ln (\frac{x^{3}}{x^{2} + x + 1})$