Problem 3 - Calculus for Biology and Medicine

Question

In Problems \(1-8\), find the gradient of each function. $$ f(x, y)=\sqrt{x^{3}-3 x y} $$

Step-by-Step Solution

Answer

The gradient is \(\nabla f(x, y) = \left(\frac{3x^2 - 3y}{2\sqrt{x^3 - 3xy}}, \frac{-3x}{2\sqrt{x^3 - 3xy}}\right)\).

Step 1: Understand the Gradient

The gradient of a function, \(abla f(x, y)\), is a vector of partial derivatives of the function with respect to its variables. For a function \(f(x, y)\), the gradient is \( abla f(x, y) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)\).

Step 2: Compute Partial Derivative with respect to x

First, find the partial derivative of \(f(x, y) = \sqrt{x^3 - 3xy}\) with respect to \(x\). Use the chain rule: \( \frac{\partial f}{\partial x} = \frac{1}{2\sqrt{x^3 - 3xy}} \cdot (3x^2 - 3y)\).

Step 3: Compute Partial Derivative with respect to y

Next, find the partial derivative of \(f(x, y) = \sqrt{x^3 - 3xy}\) with respect to \(y\). Again, apply the chain rule: \( \frac{\partial f}{\partial y} = \frac{1}{2\sqrt{x^3 - 3xy}} \cdot (-3x)\).

Step 4: Compile the Gradient

Combine both partial derivatives into a gradient vector: \(abla f(x, y) = \left(\frac{3x^2 - 3y}{2\sqrt{x^3 - 3xy}}, \frac{-3x}{2\sqrt{x^3 - 3xy}}\right)\).

Key Concepts

Partial Derivatives

Partial derivatives act like regular derivatives but for multivariable functions. In simpler terms, when a function depends on more than one variable, like the function \( f(x, y) = \sqrt{x^3 - 3xy}\), you can wonder how the function changes as one of the variables changes, while others stay constant.
To find the partial derivative with respect to \( x \), you differentiate the function treating \( y \) as a constant. Similarly, find the partial derivative with respect to \( y \) by treating \( x \) as a constant.

• Partial derivative with respect to \( x \): How \( f \) changes when \( x \) changes, \( y \) stays the same.
• Partial derivative with respect to \( y \): How \( f \) changes when \( y \) changes, \( x \) stays the same.

Partial derivatives are key in finding the gradient, which tells us the direction and rate of fastest increase at any point on the surface described by the function.

Chain Rule

The chain rule is a fundamental tool in calculus allowing us to differentiate compositions of functions, even in the context of partial derivatives. Think of it as a way to "chain" different pieces of differentiation together. This rule helps differentiate complex functions by breaking them down into simpler parts.
In our example, \( f(x, y) = \sqrt{x^3 - 3xy} \), to find the partial derivative of \( f \) with respect to \( x \) or \( y \), you use the chain rule:

• Start by considering the outer function, which in this case is the square root function \( \sqrt{u} \), where \( u = x^3 - 3xy \).
• The derivative of \( \sqrt{u} \) with respect to \( u \) is \( \frac{1}{2\sqrt{u}} \).
• Then multiply by the derivative of \( u \) with respect to \( x \) or \( y \), depending on which partial derivative you're calculating.

The chain rule allows us to effectively manage the complexity of combining the derivatives of different variables and layers of functions.

Vector Calculus

Vector calculus extends calculus to vector fields and is essential in understanding multiple dimensions. One of the central concepts is the gradient, an operation that takes a scalar function and turns it into a vector field. The gradient tells us both the direction and the rate of maximum increase of a function.
For the function \( f(x, y) \), the gradient \( abla f(x, y) \) is computed as a combination of the partial derivatives of \( x \) and \( y \). The result is expressed as a vector:\[ abla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \]This vector points in the direction of greatest increase of the function 'f'.

• The length of this vector denotes the rate of this increase.
• Each component of the vector corresponds to the directional derivatives along the x and y axes.

Vector calculus, and specifically the gradient, is foundational in fields like physics and engineering, where understanding how quantities change in space is essential.