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Chapter 2: Q 34. (page 222)

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

Short Answer

Expert verified

The required answer is $\frac{2}{x} + \ln 2$ .

Step by step solution

Step 1. Given information.

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

Step 2. Find the derivative of the given function.

$f (x) = \ln (x^{2} 2^{x}) f (x) = \ln (x^{2}) + \ln (2^{x}) f (x) = 2 \ln x + x \ln 2 f' (x) = \frac{d}{d x} (2 \ln x + x \ln 2) = \frac{d}{d x} (2 \ln x) + \frac{d}{d x} (x \ln 2) = \frac{2}{x} + \ln 2$

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Most popular questions from this chapter

Find the derivative of the absolute value function and piecewise defined function

$f (x) = \{\begin{cases} 3 x + 1 & i f x \leq 1 \\ x^{3} & i f x > 1 \end{cases}$

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

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises39-54

$f (x) = \frac{2}{1 - x}$

For each function graphed in Exercises 65-68, determine the values of $x$ at which $f$ fails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

For each function $f (x)$ and interval localid="1648297458718" $[a, b]$ in Exercises localid="1648297462718" $81 - 86$ , use the Intermediate Value Theorem to argue that the function must have at least one real root on localid="1648297466951" $[a, b]$ . Then apply Newton’s method to approximate that root.

localid="1648297471865" $f (x) = x^{3} - 3 x + 1, [a, b] = [1, 2]$

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Find the derivatives of the functions:f(x)=ln(x22x)

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:
$f (x) = \ln (x^{2} 2^{x})$