Question

Find all first partial derivatives of each function. \(f(x, y)=\tan ^{-1}(4 x-7 y)\)

Short answer

\( \frac{\partial f}{\partial x} = \frac{4}{1+(4x-7y)^2}, \frac{\partial f}{\partial y} = \frac{-7}{1+(4x-7y)^2} \).

Step-by-step solution

Identify the Problem

We are tasked with finding the first partial derivatives of the function \( f(x, y) = \tan^{-1}(4x - 7y) \). This means we need to compute \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).

Find the Partial Derivative with Respect to x

To find \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant. The derivative of \( \tan^{-1}(u) \) with respect to \( u \) is \( \frac{1}{1+u^2} \). Let \( u = 4x - 7y \). Thus, \( \frac{\partial f}{\partial x} = \frac{1}{1+(4x-7y)^2} \cdot 4 \) because the derivative of \( 4x \) with respect to \( x \) is 4. So, \( \frac{\partial f}{\partial x} = \frac{4}{1+(4x-7y)^2} \).

Find the Partial Derivative with Respect to y

To compute \( \frac{\partial f}{\partial y} \), treat \( x \) as constant. The derivative of \( \tan^{-1}(u) \) with respect to \( u \) is \( \frac{1}{1+u^2} \) with \( u = 4x - 7y \). Now, differentiate \( u \) with respect to \( y \): the derivative of \( -7y \) with respect to \( y \) is -7. Thus, \( \frac{\partial f}{\partial y} = \frac{1}{1+(4x-7y)^2} \cdot (-7) \).Therefore, \( \frac{\partial f}{\partial y} = \frac{-7}{1+(4x-7y)^2} \).

Conclusion

The first partial derivatives of the function are: \[ \frac{\partial f}{\partial x} = \frac{4}{1+(4x-7y)^2} \] and \[ \frac{\partial f}{\partial y} = \frac{-7}{1+(4x-7y)^2} \].

Key concepts

Multivariable Calculus

Multivariable calculus is a branch of mathematics that extends calculus concepts to functions of multiple variables. Let's start with understanding what a function of multiple variables means. In single-variable calculus, you have functions with one variable, like time or distance. But in multivariable calculus, a function can depend on two or more variables.
For example, consider the function we have: \( f(x, y) = \tan^{-1}(4x - 7y) \). Here, we have two variables: \( x \) and \( y \). Imagine if we could map the temperature in a room. It might depend on both the x and y coordinates within the room. That's kind of what a multivariable function does. It gives us an output based on multiple inputs.
In multivariable calculus, it's useful to talk about partial derivatives. Partial derivatives tell us how a function changes as we change just one variable while keeping the others constant. It's crucial for understanding how complex systems behave when you tweak one component at a time. This is particularly useful in physics, economics, engineering, and any field that involves relationships between multiple factors.
Understanding multivariable calculus helps us tackle more advanced problems and gives a better grasp of natural phenomena. Learning to find partial derivatives is a fundamental skill in this area.

Derivatives

Derivatives, as you might know from single-variable calculus, measure how a function changes as its input changes. It is the mathematical equivalent of asking, "What's the rate of change at this particular point?" In multivariable calculus, this concept extends to functions with more than one variable, leading us to partial derivatives.
When finding partial derivatives, think of them as a zoom-in on how the function changes along one dimension while holding everything else steady.
For our function \( f(x, y) = \tan^{-1}(4x - 7y) \), you compute the partial derivative with respect to \( x \) by treating \( y \) as a constant, and vice versa for \( y \). These derivatives reveal how the function's slope changes along the x-axis and the y-axis separately. Here's why partial derivatives are important:

• They help in optimizing and analyzing functions involving multiple variables.
• Partial derivatives form the backbone for concepts like gradients and Hessians, which are instrumental in fields like optimization and machine learning.

Every partial derivative offers a piece of the puzzle in understanding the complete picture of how a function behaves across its entire domain.

Inverse Trigonometric Functions

Inverse trigonometric functions are the functions that reverse the usual trigonometric operations. For example, the inverse of \( \tan \) is \( \tan^{-1} \), often written as arctan. These functions allow us to go backward from a trigonometric ratio to an angle. In our function \( f(x, y) = \tan^{-1}(4x - 7y) \), we're faced with the inverse tangent, which is a common function used in calculus.
Inverse functions can sometimes be less intuitive, so it's important to understand the basics:

• The domain of \( \tan^{-1} \) is all real numbers, and its range is \( (-\pi/2, \pi/2) \).
• When differentiating \( \tan^{-1}(u) \) with respect to \( u \), the derivative is \( \frac{1}{1+u^2} \). This rule is essential in finding partial derivatives of functions involving \( \tan^{-1} \).

These functions and their properties are not just academic curiosities; they have real-world applications. Engineers often use inverse trigonometric functions to model waveforms and angles in signal processing, and graphics programmers use them in designing realistic virtual environments.
By mastering inverse trigonometric functions and their derivatives, you enhance your mathematical toolkit, enabling you to tackle a wide variety of problems in calculus and applied math.