Problem 42 - Calculus Early Transcendentals

Question

Find the indicated partial derivatives. $$f(x, y, z)=\sqrt{\sin ^{2} x+\sin ^{2} y+\sin ^{2} z} ; \quad f_{z}(0,0, \pi / 4)$$

Step-by-Step Solution

Answer

The partial derivative \( f_z(0, 0, \frac{\pi}{4}) \) is \( \frac{\sqrt{2}}{2} \).

Step 1: Define the Function

The given function is \( f(x, y, z) = \sqrt{\sin^2 x + \sin^2 y + \sin^2 z} \). We are asked to find the partial derivative with respect to \( z \) at the point \((0,0,\frac{\pi}{4})\).

Step 2: Differentiate with Respect to z

To find \( f_z(x, y, z) \), apply the chain rule. The derivative of the outer function, \( \sqrt{u} \), with respect to \( u \) is \( \frac{1}{2\sqrt{u}} \). The derivative of the inner function, \( \sin^2 z \), with respect to \( z \) is \( 2\sin z \cos z \) or \( \sin(2z) \) using the double-angle identity. Thus:\[f_z(x, y, z) = \frac{2\sin z \cos z}{2\sqrt{\sin^2 x + \sin^2 y + \sin^2 z}} = \frac{\sin(2z)}{2\sqrt{\sin^2 x + \sin^2 y + \sin^2 z}}\]

Step 3: Evaluate the Derivative at the Given Point

Substitute \( x = 0 \), \( y = 0 \), and \( z = \frac{\pi}{4} \) into the derivative:1. \( \sin^2 0 + \sin^2 0 + \sin^2 \frac{\pi}{4} = 0 + 0 + \left( \frac{\sqrt{2}}{2} \right)^2 = \frac{1}{2} \).2. \( \sin(2\times\frac{\pi}{4}) = \sin(\frac{\pi}{2}) = 1 \).Thus:\[\f_z(0, 0, \frac{\pi}{4}) = \frac{1}{2\sqrt{\frac{1}{2}}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\\]

Step 4: Verify Calculations

Double-check each calculation for correctness. Confirming that:- \(\sin^2 0 = 0\)- \(\sin^2 0 = 0\)- \(\sin^2 \frac{\pi}{4} = \frac{1}{2}\)- \(\sin(\frac{\pi}{2}) = 1\)This confirms the calculations are accurate.

Key Concepts

Chain Rule

The chain rule is a powerful tool in calculus used to find the derivative of composite functions. In essence, it allows us to differentiate functions nested within each other. For a function composed of two functions, say \( f(g(x)) \), the chain rule states that the derivative is the product of the derivative of the outer function \( f' \) evaluated at \( g(x) \) and the derivative of the inner function \( g' \) with respect to \( x \). In formula terms, it is: \[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)\]This becomes especially handy when dealing with more complex, multivariable functions, such as when calculating partial derivatives. In the given exercise, the chain rule was applied to the outer function \( \sqrt{u} \), with \( u = \sin^2 x + \sin^2 y + \sin^2 z \). The chain rule required differentiating the outer function with respect to \( u \), and the inner function with respect to \( z \).

Trigonometric Identities

Trigonometric identities simplify the process of differentiating trig functions. They are formulas involving the angles and lengths of triangles that are always true for specific relationships between angles and sides. In this exercise, the double-angle identity was used, specifically:

• \( \sin(2z) = 2\sin(z)\cos(z) \)

This particular identity turns products of sine and cosine terms into a single sine function with twice the angle, simplifying differentiation. Applying this identity helped transform the derivative of \( \sin^2 z \) into \( \sin(2z) \), making it easier to calculate with the chain rule. Recognizing such identities can greatly simplify finding derivatives and integrals with trigonometric functions.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of several variables, involving more intricate concepts like partial derivatives. Partial derivatives measure how a function changes as one variable changes, keeping the others constant.
In this problem, the function \( f(x, y, z) = \sqrt{\sin^2 x + \sin^2 y + \sin^2 z} \) has three variables. To find \( f_z \), the partial derivative with respect to \( z \), we viewed \( x \) and \( y \) as constants, focusing only on the changes in \( z \).
The steps involved differentiating only with respect to \( z \), leveraging powerful calculus tools like the chain rule and trigonometric identities to simplify our work. Understanding these principles in multivariable contexts allows deeper exploration into complex systems, illustrating how small changes in one variable can affect entire functions.