Problem 17 - Calculus for Biology and Medicine

Question

Find the indicated partial derivatives. \(f(x, y)=3 x^{2}-y+2 y^{2} ; f_{x}(1,0)\)

Step-by-Step Solution

Answer

The partial derivative \( f_x(1,0) \) is 6.

Step 1: Identify the Function

We are given the function \( f(x, y) = 3x^2 - y + 2y^2 \) and we are tasked with finding the partial derivative with respect to \( x \) at the point \( (1, 0) \).

Step 2: Apply Partial Derivative with Respect to x

To find the partial derivative \( f_x(x, y) \), differentiate the function \( f(x, y) \) with respect to \( x \), treating \( y \) as a constant. The derivative of \( 3x^2 \) with respect to \( x \) is \( 6x \). The terms \( -y \) and \( 2y^2 \) are constant with respect to \( x \), so their derivatives are both zero. Thus, \( f_x(x, y) = 6x \).

Step 3: Evaluate the Partial Derivative at the Given Point

Now that we have \( f_x(x, y) = 6x \), we evaluate this at the point \( (1, 0) \). Substitute \( x = 1 \) into \( f_x(x, y) = 6x \), giving us \( f_x(1, 0) = 6(1) = 6 \).

Key Concepts

Multivariable Calculus

In mathematics, multivariable calculus extends the concepts of calculus to functions of several variables. Unlike single-variable functions, which have only one independent variable, multivariable functions depend on two or more variables. This field is essential for analyzing systems in physics, engineering, economics, and more, where change occurs with respect to multiple factors.

An example of a multivariable function is the given function \( f(x, y) = 3x^2 - y + 2y^2 \). This function has two independent variables, \( x \) and \( y \), allowing it to model phenomena where outcomes depend on these two inputs.

When dealing with such functions, we often want to understand how changes in these inputs affect the output. Partial derivatives help us pinpoint how the function changes as each variable varies independently, holding the others constant.

Derivative Techniques

In the realm of multivariable calculus, partial derivatives are a primary derivative technique. They measure the rate at which a function changes as one particular variable changes, while other variables are kept constant. This technique is especially useful in scenarios where functions describe surfaces or curves in three-dimensional space.

To find the partial derivative of \( f(x, y) = 3x^2 - y + 2y^2 \) with respect to \( x \), we focus solely on how \( x \) affects \( f(x, y) \). The steps are:

• Differentiate \( 3x^2 \) with respect to \( x \), which gives \( 6x \).
• Treat \( -y \) and \( 2y^2 \) as constants since we are deriving with respect to \( x \). Their derivatives are zero.

Thus, we find that the partial derivative of \( f \) with respect to \( x \) is \( 6x \). This result shows how rapidly \( f \) changes as \( x \) changes, holding \( y \) fixed.

Function Differentiation

Function differentiation involves finding the derivative of a functional expression. In our context, it means computing the partial derivatives for each independent variable in a multivariable function.

When you have a function like \( f(x, y) = 3x^2 - y + 2y^2 \), differentiating with respect to \( x \) while treating \( y \) as constant isolates the effect of \( x \) on the function's output.

For the exercise, after computing the partial derivative \( f_x(x, y) = 6x \), we then evaluate it by substituting \( x = 1 \). Calculating this derivative at point \( (1, 0) \) yields \( f_x(1, 0) = 6(1) = 6 \). This specific value gives the slope of the tangent to the curve at that point, reflecting how sensitive the function is to changes in \( x \) at \( x = 1 \), while \( y \) is held constant at 0.