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Chapter 2: Q. 22 (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.
$f (x) = {(\sqrt{x} + 4)}^{5}$

Short Answer

Expert verified

The required answer is $\frac{5 {(\sqrt{x} + 4)}^{4}}{2 \sqrt{x}}$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = {(\sqrt{x} + 4)}^{5}$

02

Step 2. Calculation

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

$f' (x) = \frac{d}{d x} {(\sqrt{x} + 4)}^{5} = 5 {(\sqrt{x} + 4)}^{5 - 1} \frac{d}{d x} (\sqrt{x} + 4) = 5 {(\sqrt{x} + 4)}^{4} (\frac{d}{d x} \sqrt{x} + \frac{d}{d x} 4) = 5 {(\sqrt{x} + 4)}^{4} (\frac{1}{2} x^{\frac{1}{2} - 1}) = 5 {(\sqrt{x} + 4)}^{4} (\frac{1}{2} x^{- \frac{1}{2}}) = \frac{5 {(\sqrt{x} + 4)}^{4}}{2 \sqrt{x}}$

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Most popular questions from this chapter

Taking the limit: We have seen that if f is a smooth function, then $f^{'} (c) \approx \frac{f (c + h) - f (c)}{h}$ This approximation should get better as h gets closer to zero. In fact, in the next section we will define the derivative in terms of such a limit.

$f^{'} (c) = \lim_{h \to 0} \frac{f (c + h) - f (c)}{h}$ .

Use the limit just defined to calculate the exact slope of the tangent line to $f (x) = x^{2}$ at $x = 4 .$

Stuart left his house at noon and walked north on Pine Street for $20$ minutes. At that point he realized he was late for an appointment at the dentist, whose office was located south of Stuart’s house on Pine Street; fearing he would be late, Stuart sprinted south on Pine Street, past his house, and on to the dentist’s office. When he got there, he found the office closed for lunch; he was $10$ minutes early for his $12 : 40$ appointment. Stuart waited at the office for $10$ minutes and then found out that his appointment was actually for the next day, so he walked back to his house. Sketch a graph that describes Stuart’s position over time. Then sketch a graph that describes Stuart’s velocity over time.

Suppose f is a polynomial of degree n and let k be some integer with $0 \leq k \leq n$ . Prove that if f(x) is of the form $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . . + a_{1} x + a_{0}$

Then $a_{k} = \frac{f^{k} (0)}{k!}$ where $f^{k} (x)$ is the k-th derivative of $f (x) a n d k! = k \cdot (k - 1) \cdot (k - 2) . . . . 2 \cdot 1$

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.

$f (x) = \frac{1}{\sqrt{x^{2} + 1}}$

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 34-59

$f (x) = 3 \sqrt{x}$

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