Question

Suppose \(F(x, y)=0\) and \(y\) is a differentiable function of \(x\) Explain how to find \(d y / d x\).

Short answer

Question: Given a function F(x, y) = 0, where y is a differentiable function of x, find the derivative of y with respect to x, i.e., \(dy/dx\). Answer: \(dy/dx = -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}\)

Step-by-step solution

Find the total differential of F(x, y)

To find the total differential of F(x, y) = 0, we will differentiate the function with respect to x and y. We do this using the chain rule, which states that: $$ \frac{dF}{dx} = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx} $$ The left-hand side (\(dF/dx\)) is equal to zero since F(x, y) is a constant function.

Calculate the partial derivatives of F(x, y)

Next, we need to calculate the partial derivatives \(\partial F/\partial x\) and \(\partial F/\partial y\). Since we are not given the explicit form of F(x, y), we cannot compute these partial derivatives directly. However, we can represent them symbolically with the notation \(\frac{\partial F}{\partial x}\) and \(\frac{\partial F}{\partial y}\).

Substitute the partial derivatives into the total differential equation

Now, let's substitute the partial derivatives into the total differential equation from Step 1: $$ 0 = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx} $$

Solve the equation for \(dy/dx\)

Finally, we need to solve the equation for \(dy/dx\). We can isolate the \(dy/dx\) term by moving the \(\frac{\partial F}{\partial x}\) term to the other side of the equation: $$ \frac{dy}{dx} = -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}} $$ Now, we have found the derivative of y with respect to x, expressed in terms of the partial derivatives of the function F(x, y). To find the derivative for a specific function F(x, y), we would need to calculate the partial derivatives \(\frac{\partial F}{\partial x}\) and \(\frac{\partial F}{\partial y}\) from that specific function and substitute them into our solution.

Key concepts

Chain Rule

When we look at functions that are composed of other functions, we can't use the ordinary rules of differentiation. This is where the chain rule comes into play. Imagine a scenario where we have a function, say, a sphere’s volume that depends on its radius, but the radius is changing over time. To find out how the volume changes over time, we must consider how the radius changes with respect to time.

The chain rule helps us to differentiate composite functions by breaking down the function into its inner and outer parts. For each part, we differentiate and then multiply these derivatives together. The formula for the chain rule is given by: \[ \frac{d}{dx}(f(g(x))) = f'(g(x)) \times g'(x) \].

By using the chain rule, you can find how changes in one variable affect another variable indirectly, through an intermediary variable or function.

Total Differential

The concept of total differential is crucial when dealing with functions of multiple variables. It generalizes the derivative to functions of several variables. The total differential gives us a way to approximate how much the function's value changes in response to small changes in each variable.

The total differential of a function, for example, temperature variation across a surface over time, is expressed as: \[ dT = \frac{\partial T}{\partial x}dx + \frac{\partial T}{\partial y}dy + \frac{\partial T}{\partial t}dt \].

In this expression, the partial derivatives represent the rates of change of the temperature with respect to each individual variable, while preserving the other variables constant. The infinitesimal increases in each variable are represented by dx, dy, and dt.

Partial Derivatives

Partial derivatives are an extension of the concept of derivatives for functions of more than one variable. When we have a function that depends on several variables, and we are interested in finding out how that function changes with respect to one of those variables, we turn to partial derivatives.

Mathematically, a partial derivative of a function with respect to a variable is the derivative of the function while keeping other variables constant. For instance, if we have a function \( z = f(x, y) \), the partial derivative of \( z \) with respect to \( x \) is denoted by \( \frac{\partial z}{\partial x} \) and calculated by treating \( y \) as a constant.

A real-world example could be the rate at which the pressure inside a balloon changes with respect to the temperature, assuming the volume is held constant.

Derivative of a Function

The derivative of a function represents the rate at which the output value of the function changes as its input value changes. In the context of single-variable calculus, the derivative tells you how steep the graph of the function is at any point. If we have a function \( y = f(x) \), the derivative of the function at any point \( x \) is denoted as \( \frac{dy}{dx} \) or \( f'(x) \).

Derivatives have practical applications in various scientific fields, including physics, where they can describe velocities and accelerations, and economics, where they can measure rates of change in cost and revenue. Understanding the concept of derivatives enables one to predict and model the behavior of dynamic systems effectively.