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Chapter 3: Problem 8

Find all second partial derivatives of \(f(x, y)=\sec ^{3}(4 y-3 x)\).

Short Answer

Expert verified
The second partial derivatives are \(f_{xx}(x, y)\), \(f_{yy}(x, y)\), \(f_{xy}(x, y)\), and \(f_{yx}(x, y)\).

Step by step solution

01

Find the first partial derivative with respect to x

To find the first partial derivative of the function \( f(x, y) = \sec^3(4y - 3x) \) with respect to \( x \), apply the chain rule. First, note that the derivative of \( \sec^3(u) \) with respect to \( u \) is \( 3\sec^2(u) \cdot \sec(u)\tan(u) \). Therefore, for \( f(x, y) \), \( u = 4y - 3x \). Then, take the derivative of the inside function \(-3\) with respect to \( x \):\[ f_x(x, y) = 3\sec^2(4y - 3x) \cdot \sec(4y - 3x)\tan(4y - 3x) \cdot (-3) \]\[ f_x(x, y) = -9\sec^3(4y - 3x)\tan(4y - 3x) \].
02

Find the first partial derivative with respect to y

Similarly, to find the first partial derivative with respect to \( y \), apply the chain rule. The outside function again is \( \sec^3(u) \), with the same derivative as before. The derivative of the inside function \( 4y - 3x \) with respect to \( y \) is \( 4 \):\[ f_y(x, y) = 3\sec^2(4y - 3x) \cdot \sec(4y - 3x)\tan(4y - 3x) \cdot 4 \]\[ f_y(x, y) = 12\sec^3(4y - 3x)\tan(4y - 3x) \].
03

Find the second partial derivative with respect to x

To find the second partial derivative \( f_{xx}(x, y) \), differentiate \( f_x(x, y) = -9\sec^3(4y - 3x)\tan(4y - 3x) \) with respect to \( x \). Apply the product rule along with the chain rule. The derivative of \( \tan(v) \) is \( \sec^2(v) \). Consider also the derivative of the inside function \(-3\):\[ f_{xx}(x, y) = -9 \left[ 3\sec^2(4y - 3x) \cdot \tan(4y - 3x) (-3) + \sec^3(4y - 3x) \cdot \sec^2(4y - 3x)(-3)\right] \]This expands to:\[ f_{xx}(x, y) = -27\sec^3(4y - 3x)\tan^2(4y - 3x) - 27\sec^5(4y - 3x) \].
04

Find the second partial derivative with respect to y

Now, differentiate \( f_y(x, y) = 12\sec^3(4y - 3x)\tan(4y - 3x) \) with respect to \( y \) using the product rule and chain rule:\[ f_{yy}(x, y) = 12 \left[ 3\sec^2(4y - 3x) \tan(4y - 3x) (4) + \sec^3(4y - 3x) \cdot \sec^2(4y - 3x) (4) \right ] \]This yields:\[ f_{yy}(x, y) = 144\sec^3(4y - 3x)\tan^2(4y - 3x) + 144\sec^5(4y - 3x) \].
05

Find the mixed partial derivative with respect to x and then y

For the mixed partial \( f_{xy}(x, y) \), differentiate the first partial derivative with respect to \( x \) (\( f_x(x, y) = -9\sec^3(4y - 3x)\tan(4y - 3x) \)) with respect to \( y \). Apply the product and chain rules:\[ f_{xy}(x, y) = -9 \left[ 3\sec^2(4y - 3x) \tan(4y - 3x) (4) + \sec^3(4y - 3x) \cdot \sec^2(4y - 3x)(4)\right] \]Simplifying, we get:\[ f_{xy}(x, y) = -108\sec^3(4y - 3x)\tan^2(4y - 3x) - 108\sec^5(4y - 3x) \].
06

Find the mixed partial derivative with respect to y and then x

For the mixed derivative \( f_{yx}(x, y) \), differentiate \( f_y(x, y) = 12\sec^3(4y - 3x)\tan(4y - 3x) \) with respect to \( x \):\[ f_{yx}(x, y) = 12 \left[ 3\sec^2(4y - 3x) \tan(4y - 3x) (-3) + \sec^3(4y - 3x) \cdot \sec^2(4y - 3x)(-3)\right] \]Simplifying, we obtain:\[ f_{yx}(x, y) = -108\sec^3(4y - 3x)\tan^2(4y - 3x) - 108\sec^5(4y - 3x) \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chain Rule
The Chain Rule is a fundamental concept in calculus used for finding derivatives of compositions of functions. When you have a function inside another function, like in the case of \( f(x, y) = \sec^3(4y - 3x) \), the Chain Rule helps differentiate these nested functions.

Let's break it down step-by-step:
  • Identify the outside and inside functions. Here, the outside function is \( \sec^3(u) \), and the inside function is \( u = 4y - 3x \).
  • Differentiating the outside function: If \( v = \sec^3(u) \), then \( dv/du = 3\sec^2(u) \cdot \sec(u)\tan(u) \).
  • Differentiating the inside function: For example, with respect to \( x \), we get \( du/dx = -3 \), and with respect to \( y \), we have \( du/dy = 4 \).
  • Apply the Chain Rule: Multiply the derivatives from the chain. For \( f_x(x, y) \), this would be \( 3\sec^2(4y - 3x) \cdot \sec(4y - 3x)\tan(4y - 3x) \cdot (-3) \).
This structured approach ensures you accurately compute derivatives involving nested functions.
Product Rule
The Product Rule is crucial when differentiating functions that are multiplied together. When handling a function like the partial derivative, where we have terms involving products such as \( \sec^3(4y - 3x) \cdot \tan(4y - 3x) \), the Product Rule comes into play.

Here鈥檚 how you can think about it:
  • The Product Rule states: If \( u(x, y) \) and \( v(x, y) \) are functions, then \( (uv)' = u'v + uv' \).
  • Apply this to find second partial derivatives. For instance, differentiating \( f_x(x, y) = -9\sec^3(4y - 3x)\tan(4y - 3x) \) with respect to \( x \), you treat \( \sec^3(4y - 3x) \) as one function and \( \tan(4y - 3x) \) as another.
  • Compute the derivative of each separately, and then use the rule: \( [\sec^3(4y - 3x)]' \) and \( [\tan(4y - 3x)]' \).
By applying the Product Rule correctly, you can break down complex derivatives into manageable parts, yielding accurate results.
Partial Derivatives
Partial derivatives extend the concept of derivatives to functions of multiple variables. They help us understand how a function behaves with respect to each individual variable, holding the others constant.

Some key points include:
  • When given a function \( f(x, y) \), like \( f(x, y) = \sec^3(4y - 3x) \), a partial derivative in respect to \( x \), noted as \( f_x(x, y) \), measures how \( f \) changes as \( x \) changes while \( y \) remains constant.
  • Similarly, the partial derivative with respect to \( y \), noted \( f_y(x, y) \), measures how \( f \) changes as \( y \) changes while \( x \) remains constant.
  • For second partial derivatives, you determine the "rate of change" of the first partial derivatives, such as \( f_{xx}(x, y) \) and \( f_{yy}(x, y) \).
  • Mixed partial derivatives, like \( f_{xy}(x, y) \) and \( f_{yx}(x, y) \), evaluate how the function changes with respect to two variables in succession.
By understanding partial derivatives, you gain deeper insights into the behavior and curvature of multivariable functions.

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