Q. 50 Use the differentiation rules de... [FREE SOLUTION]

Chapter 2: Q. 50 (page 198)

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.
$f (x) = {(\sqrt{x} - \sqrt[3]{x})}^{2}$

Short Answer

Expert verified

The derivative of the function is $f' (x) = \begin{array}{rcl} 1 \end{array} \begin{array}{rcl} - \end{array} \begin{array}{rcl} \frac{5}{3} \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} \end{array}$ ^-16+23x^-13.

Step by step solution

Step 1. Given Information

The given function is $f (x) = {(\sqrt{x} - \sqrt[3]{x})}^{2}$ .

Step 2. Simplify the function

Apply the identity ${(a - b)}^{2} = a^{2} - 2 a b + b^{2}$ in the given function.

role="math" localid="1648549066913" $\begin{array}{rcl} f (x) & = & {(\sqrt{x})}^{2} - 2 \sqrt{x} \sqrt[3]{x} + {(\sqrt[3]{x})}^{2} = x - 2 x^{\frac{1}{2} + \frac{1}{3}} + x^{\frac{2}{3}} \\ = & x - 2 x^{\frac{5}{6}} + x^{\frac{2}{3}} \end{array}$

Step 3. Find the derivative

Apply the sum rule of derivative, $(f + g)' (x) = f' (x) + g' (x)$ and the difference rule of derivative, $(f - g)' (x) = f' (x) - g' (x)$ .

localid="1648549080637" $f' (x) = \frac{d}{d x} (\begin{array}{rcl} x \end{array} \begin{array}{rcl} ) \end{array} \begin{array}{rcl} - \end{array} \begin{array}{rcl} \frac{d}{d x} \end{array} \begin{array}{rcl} ( \end{array} \begin{array}{rcl} 2 \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} ^{\frac{5}{6}} \end{array} \begin{array}{rcl} ) \end{array} \begin{array}{rcl} + \end{array} \begin{array}{rcl} \frac{d}{d x} \end{array} \begin{array}{rcl} ( \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} ^{\frac{2}{3}} \end{array} \begin{array}{rcl} ) \end{array}$

Apply the constant multiple rule of derivative, $(k f)' (x) = k f' (x)$ .

localid="1648549090521" $f' (x) = \frac{d}{d x} (\begin{array}{rcl} x \end{array} \begin{array}{rcl} ) \end{array} \begin{array}{rcl} - \end{array} \begin{array}{rcl} 2 \frac{d}{d x} \end{array} \begin{array}{rcl} ( \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} ^{\frac{5}{6}} \end{array} \begin{array}{rcl} ) \end{array} \begin{array}{rcl} + \end{array} \begin{array}{rcl} \frac{d}{d x} \end{array} \begin{array}{rcl} ( \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} ^{\frac{2}{3}} \end{array} \begin{array}{rcl} ) \end{array}$

Apply the power rule of derivative, localid="1648555968108" $(x^{n})' = n x^{n - 1}$ .

localid="1648549175793" $\begin{array}{rcl} f' (x) & = & (1) - 2 (\frac{5}{6} x^{\frac{- 1}{6}}) + (\frac{2}{3} x^{\frac{- 1}{3}}) \\ = & 1 - \frac{5}{3} x^{\frac{- 1}{6}} + \frac{2}{3} x^{\frac{- 1}{3}} \end{array}$

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Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.
$f (x) = {(\sqrt{x} - \sqrt[3]{x})}^{2}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Simplify the function

Step 3. Find the derivative

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91影视

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating. f(x)=(x-x3)2

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Simplify the function

Step 3. Find the derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Statistics

Mechanics Maths

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.
$f (x) = {(\sqrt{x} - \sqrt[3]{x})}^{2}$