Chapter 10: Q. 5 (page 777)

Use the definition of the derivative to calculate the derivative of $f (x) = \sqrt{x}$ at $c = 4$ . At some point you will need to multiply numerator and denominator by the conjugate of $\sqrt{4 + h} - 2$ , which is $\sqrt{4 + h} + 2$ .

Short Answer

Expert verified

The derivative is $\frac{1}{4}$ .

Step by step solution

Step 1. Given Information.

We are given a function,

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

Step 2. Finding the derivative.

The derivative is given by,

$f^{'} (c) = \lim_{h \to 0} \frac{f (c + h) - f (c)}{h} f^{'} (4) = \lim_{h \to 0} \frac{\sqrt{4 + h} - \sqrt{4}}{h} = \lim_{h \to 0} \frac{\sqrt{4 + h} - 2}{h} \times \frac{\sqrt{4 + h} + 2}{\sqrt{4 + h} + 2} = \lim_{h \to 0} \frac{4 + h - 4}{h (\sqrt{4 + h} + 2)} = \lim_{h \to 0} \frac{h}{h (\sqrt{4 + h} + 2)} = \lim_{h \to 0} \frac{1}{\sqrt{4 + h} + 2} = \frac{1}{\sqrt{4} + 2} = \frac{1}{4}$

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Use the definition of the derivative to calculate the derivative of $f (x) = \sqrt{x}$ at $c = 4$ . At some point you will need to multiply numerator and denominator by the conjugate of $\sqrt{4 + h} - 2$ , which is $\sqrt{4 + h} + 2$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the derivative.

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Use the definition of the derivative to calculate the derivative of f(x)=x at c=4. At some point you will need to multiply numerator and denominator by the conjugate of 4+h-2, which is 4+h+2.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the derivative.

One App. One Place for Learning.

Most popular questions from this chapter

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Use the definition of the derivative to calculate the derivative of $f (x) = \sqrt{x}$ at $c = 4$ . At some point you will need to multiply numerator and denominator by the conjugate of $\sqrt{4 + h} - 2$ , which is $\sqrt{4 + h} + 2$ .