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Chapter 2: Q 35. (page 222)

Find the derivatives of the functions: $f (x) = x^{2} \log_{2} (x 2^{x})$

Short Answer

Expert verified

The required answer is $3 x^{2} + 2 x \log_{2} x + \frac{x}{\ln 2}$

Step by step solution

01

Step 1. Given information.

The given function is: $f (x) = x^{2} \log_{2} (x 2^{x})$

02

Step 2. Find the derivative of the given function.

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

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Most popular questions from this chapter

In Exercises 69鈥�80, determine whether or not $f$ is continuous and/or differentiable at the given value of $x$ . If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

$f (x) = \{\begin{cases} x^{2}, i f x 鈮� 1 \\ 2 x - 1, i f x > 1, \end{cases} x = 1$

Use the limit you just found to calculate the exact slope of the tangent line to $f (x) = x^{2}$ at $x = 4$ . Obviously you should get the same final answer as you did earlier.

In the text we noted that if $f (u (v (x)))$ was a composition of three functions, then its derivative is $\frac{d f}{d x} = \frac{d f}{d u} 脳 \frac{d u}{d v} 脳 \frac{d v}{d x}$ . Write this rule in 鈥減rime鈥� notation.

Each graph in Exercises 31鈥�34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

Use the definition of the derivative to find $f$ for each function $f$ in Exercises 39-54

$f (x) = \frac{x^{3}}{x + 1}$

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