Chapter 12: Chapter 12

Problem 10

Calculate \(\frac{d^{2} y}{d x^{2}}\). . \(y=x^{-2}+\ln x\)

Short Answer

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The second derivative of the given function, expressed as \(\frac{d^2 y}{d x^2}\), is: \[\frac{d^2 y}{d x^2} = \frac{6}{x^4} - \frac{1}{x^2}.\]

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Differentiate the given function y with respect to x

To find the first derivative of \(y = x^{-2} + \ln x\), we will differentiate each term with respect to x. For the first term, \(x^{-2}\), we apply the power rule for differentiation: \(\frac{d}{dx}(x^n) = nx^{n-1}\). Thus, \[\frac{d}{dx}(x^{-2}) = -2x^{-3}.\] For the second term, \(\ln x\), the derivative of the natural logarithmic function is: \[\frac{d}{dx}(\ln x) = \frac{1}{x}.\] Now, adding both derivative terms, we find the first derivative, \(y'\): \[y' = -2x^{-3} + \frac{1}{x}.\]

Step 2: Differentiate the 1st derivative for 2nd derivative

Having found the first derivative of the function, we will now differentiate it to find the second derivative \(\frac{d^2 y}{d x^2}\). Differentiating each term separately, we get: For the first term, \(-2x^{-3}\), we apply the power rule again: \[\frac{d}{dx}(-2x^{-3}) = 6x^{-4}.\] For the second term, \(\frac{1}{x}\), the derivative is: \[\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}.\] Now, adding both derivative terms, we find the second derivative, \(y''\): \[y'' = 6x^{-4} - \frac{1}{x^2}.\]

Step 3: Write the final answer in a simplified form

Finally, to express the second derivative in a more simplified form, we can rewrite the \(x^{-4}\) term as \(\frac{1}{x^4}\): \[y'' = 6\frac{1}{x^4} - \frac{1}{x^2}.\] So, the second derivative of the given function, expressed as \(\frac{d^2 y}{d x^2}\), is: \[\frac{d^2 y}{d x^2} = \frac{6}{x^4} - \frac{1}{x^2}.\]

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Chapter 12: Chapter 12

Calculate \(\frac{d^{2} y}{d x^{2}}\). . \(y=x^{-2}+\ln x\)

Short Answer

Step by step solution

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Step 1: Differentiate the given function y with respect to x

Step 2: Differentiate the 1st derivative for 2nd derivative

Step 3: Write the final answer in a simplified form

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More chapters from the book ‘Finite Mathematics and Applied Calculus’

Chapter 0

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 13

Chapter 14

Chapter 15

Chapter 16

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