Q. 71 Let聽z=(聽f(x,聽y))n.聽Show聽tha... [FREE SOLUTION]

Chapter 12: Q. 71 (page 945)

$Let z = (f {(x, y))}^{n} . Show that \frac{鈭� z}{鈭� x} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� x} and \frac{鈭� z}{鈭� y} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� y} .$

Short Answer

Expert verified

$\frac{鈭� z}{鈭� x} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� x} \frac{鈭� z}{鈭� y} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� y}$

Step by step solution

Step 1. Given information

A function, $z = (f {(x, y))}^{n}$

Step 2. Proofs of given partial derivatives

$LHS = \frac{鈭� z}{鈭� x} = \frac{鈭� (f {(x, y))}^{n}}{鈭� x} Using power rule, chain rule and definition of partial derivatives, it follows that : = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� x} = RHS Similarly, for partial derivative with respect to y, we get, LHS = \frac{鈭� z}{鈭� y} = \frac{鈭� (f {(x, y))}^{n}}{鈭� y} Using power rule, chain rule and definition of partial derivatives, it follows that : = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� y} = RHS$

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91影视

$Let z = (f {(x, y))}^{n} . Show that \frac{鈭� z}{鈭� x} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� x} and \frac{鈭� z}{鈭� y} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� y} .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proofs of given partial derivatives

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91影视

Letz=(f(x,y))n.Showthat鈭�z鈭�x=n(f(x,y))n-1鈭�f鈭�xand鈭�z鈭�y=n(f(x,y))n-1鈭�f鈭�y.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proofs of given partial derivatives

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Applied Mathematics

Geometry

Discrete Mathematics

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

$Let z = (f {(x, y))}^{n} . Show that \frac{鈭� z}{鈭� x} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� x} and \frac{鈭� z}{鈭� y} = n (f {(x, y))}^{n - 1} \frac{鈭� f}{鈭� y} .$