Chapter 2: Q. 42 (page 210)

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.
$f (x) = x \sqrt{3 x^{2} + 1} \sqrt[3]{2 x + 5}$

Short Answer

Expert verified

The required answer is $\sqrt{3 x^{2} + 1} \sqrt[3]{2 x + 5} + \frac{3 x^{2} \sqrt[3]{2 x + 5}}{\sqrt{3 x^{2} + 1}} + 3 x \sqrt{3 x^{2} + 1} \sqrt[3]{2 x + 5}$

Step by step solution

Step 1. Given Information

The given function is $f (x) = x \sqrt{3 x^{2} + 1} \sqrt[3]{2 x + 5}$

Step 2. Calculation

Differentiate both the sides with respect to x, we get,

$f' (x) = (\frac{d}{d x} x) \sqrt{3 x^{2} + 1} \sqrt[3]{2 x + 5} + x (\frac{d}{d x} \sqrt{3 x^{2} + 1}) \sqrt[3]{2 x + 5} + x \sqrt{3 x^{2} + 1} (\frac{d}{d x} \sqrt[3]{2 x + 5}) = \sqrt{3 x^{2} + 1} \sqrt[3]{2 x + 5} + x (\frac{1}{2} {(3 x^{2} + 1)}^{\frac{1}{2} - 1} \frac{d}{d x} (3 x^{2} + 1)) \sqrt[3]{2 x + 5} + x \sqrt{3 x^{2} + 1} (\frac{3}{2} {(2 x + 5)}^{\frac{3}{2} - 1} \frac{d}{d x} (2 x + 5)) = \sqrt{3 x^{2} + 1} \sqrt[3]{2 x + 5} + x (3 x {(3 x^{2} + 1)}^{\frac{- 1}{2}}) \sqrt[3]{2 x + 5} + x \sqrt{3 x^{2} + 1} (3 {(2 x + 5)}^{\frac{1}{2}}) = \sqrt{3 x^{2} + 1} \sqrt[3]{2 x + 5} + \frac{3 x^{2} \sqrt[3]{2 x + 5}}{\sqrt{3 x^{2} + 1}} + 3 x \sqrt{3 x^{2} + 1} \sqrt[3]{2 x + 5}$

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Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.
$f (x) = x \sqrt{3 x^{2} + 1} \sqrt[3]{2 x + 5}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

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Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating. f(x)=x3x2+12x+53

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

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 in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.
$f (x) = x \sqrt{3 x^{2} + 1} \sqrt[3]{2 x + 5}$