Chapter 2: Q. 66 (page 234)

Find the function from its given derivative
$f^{'} (x) = \frac{2 x}{\sqrt{1 - 4 x^{2}}}$

Short Answer

Expert verified

The function from its given derivative is

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

Step by step solution

Step 1.Given information

WE have been given the derivative of function

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

Step 2.We have been finding the function whose derivative is given

One can suspect the $\sin^{- 1} x$ from the denominator of function

\begin{aligned} f (x) = \sin^{- 1} (2 x) \\ f (x) = \sin^{- 1} (2 x) \\ f^{'} (x) = \frac{1}{\sqrt{1 - (2 x)^{2}}} \frac{d}{d x} (2 x) \\ = \frac{2}{\sqrt{1 - 4 x^{2}}} \end{aligned}

Step 3.Finding the more appropriate function from previous one

Now , letting the function $f (x) = - \frac{1}{2} \sqrt{1 - 4 x^{2}}$

\begin{aligned} f (x) = - \frac{1}{2} \sqrt{1 - 4 x^{2}} \\ f^{'} (x) = - \frac{1}{2} \frac{d}{d x} {(1 - 4 x^{2})}^{\frac{1}{2}} \\ = - (\frac{1}{2}) \frac{1}{2} {(1 - 4 x^{2})}^{- \frac{1}{2}} \frac{d}{d x} (1 - 4 x^{2}) [\frac{d}{d x} x^{n} = n x^{n - 1}] \\ = - \frac{1}{4 \sqrt{1 - 4 x^{2}}} (\frac{d}{d x} 1 - \frac{d}{d x} 4 x^{2}) \\ = \frac{1}{A \sqrt{1 - 4 x^{2}}} (A (2 x)) \\ = - \frac{- 2 x}{\sqrt{1 - 4 x^{2}}} \\ = \frac{2 x}{\sqrt{1 - 4 x^{2}}} \end{aligned}

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Find the function from its given derivative
$f^{'} (x) = \frac{2 x}{\sqrt{1 - 4 x^{2}}}$

Short Answer

Step by step solution

Step 1.Given information

Step 2.We have been finding the function whose derivative is given

Step 3.Finding the more appropriate function from previous one

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Find the function from its given derivativef′(x)=2x1−4x2

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Step by step solution

Step 1.Given information

Step 2.We have been finding the function whose derivative is given

Step 3.Finding the more appropriate function from previous one

One App. One Place for Learning.

Most popular questions from this chapter

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Find the function from its given derivative
$f^{'} (x) = \frac{2 x}{\sqrt{1 - 4 x^{2}}}$