Question

Find all first partial derivatives of each function. \(f(x, y)=\left(4 x-y^{2}\right)^{3 / 2}\)

Short answer

\( \frac{\partial f}{\partial x} = 6 (4x - y^2)^{1/2} \), \( \frac{\partial f}{\partial y} = -3y (4x - y^2)^{1/2} \).

Step-by-step solution

Identify the Problem

We are given the function \( f(x, y) = (4x - y^2)^{3/2} \) and asked to find its first partial derivatives with respect to \( x \) and \( y \).

Understand the Function

The function \( f(x, y) = (4x - y^2)^{3/2} \) is a composition of functions where the outer function is a power function and the inner function is \( 4x - y^2 \).

Partial Derivative with respect to x

We use the chain rule for partial differentiation. The derivative of \( (u)^{3/2} \) with respect to \( u \) is \( \frac{3}{2} u^{1/2} \). For \( u = 4x - y^2 \), its partial derivative with respect to \( x \) is \( 4 \). Thus, \( \frac{\partial f}{\partial x} = \frac{3}{2} (4x - y^2)^{1/2} \cdot 4 = 6 (4x - y^2)^{1/2} \).

Partial Derivative with respect to y

Again by using the chain rule, the derivative of \( (u)^{3/2} \) with respect to \( u \) is \( \frac{3}{2} u^{1/2} \). The partial derivative of \( u = 4x - y^2 \) with respect to \( y \) is \( -2y \). Thus, \( \frac{\partial f}{\partial y} = \frac{3}{2} (4x - y^2)^{1/2} \cdot (-2y) = -3y (4x - y^2)^{1/2} \).

Summarize the Results

The first partial derivatives are: \( \frac{\partial f}{\partial x} = 6 (4x - y^2)^{1/2} \) and \( \frac{\partial f}{\partial y} = -3y (4x - y^2)^{1/2} \).

Key concepts

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to compute the derivative of a composite function. Imagine you have a function nested inside another, kind of like a series of Russian dolls. To find the rate of change, or derivative, of the entire setup, you need to carefully peel back each layer.
The chain rule essentially says: to differentiate the composite function, you first differentiate the outer function and multiply by the derivative of the inner function.
In mathematical terms, if you have a composite function \( f(g(x)) \), the chain rule tells us that the derivative \( f'(g(x)) \cdot g'(x) \) is the way to go.

• Identify: Find the inner and outer functions.
• Differentiation: Differentiate each function respectively.
• Apply: Multiply the derivatives as per the chain rule.

Power Function

A power function is a function where a variable is raised to a constant power. They form a critical piece of many mathematical functions and are the cornerstone for concepts like polynomials.
Power functions are usually in the form \( f(x) = x^n \), where \( n \) is a constant.
When differentiating power functions, you use a specific rule known as the power rule.
The power rule states that if \( f(x) = x^n \), then its derivative \( f'(x) = n \cdot x^{n-1} \). This forms the first step in handling any differential equation involving power functions.

• Recognize the Pattern: Identify the power and apply the power law for differentiation.
• Differentiate: Lower the power by one and multiply by the original power.

Function Composition

Function composition involves creating a new function by combining two existing functions. Imagine function composition as a factory line where one operation feeds into the next.
If you have two functions, \( g(x) \) and \( f(x) \), composing them means applying one function's result directly as the input to the next, written as \( f(g(x)) \).
This layered approach is key when differentiating more complicated functions, as you will need to apply rules like the chain rule when tackling them. It allows us to break down complex problems into manageable parts.

• Define: Map one function within another.
• Differentiate: Use derivative rules like the chain rule to unpack the composition.

Calculus

Calculus is a branch of mathematics focused on the concepts of limits, derivatives, integrals, and infinite series. It's fundamentally about change: how to measure it and how to calculate it.
When discussing calculus, we especially focus on two fundamental ideas: differentiation and integration.
Differentiation refers to finding the derivative of a function, which is a way of showing the function's rate of change.
In contrast, integration deals with finding the area under the curve of a function.

• Differentiation: Unraveling how functions change.
• Integration: Summing infinitesimally small pieces to find the whole.

Calculus forms the bedrock for much of modern mathematics, allowing us to solve problems across physics, engineering, economics, and beyond.