Derivative of One Function with Respect to Another Function

In calculus, the derivative of a function measures how the function's output changes as its input changes. Typically, we think of derivatives with respect to a variable, usually denoted as $x$. However, we can also consider the derivative of one function with respect to another function. This concept is particularly useful in various applications, such as parametric equations, related rates problems, and when dealing with functions that are implicitly defined.

Understanding the Concept

When we talk about the derivative of one function with respect to another, we are interested in how one function changes as another function changes, rather than how a function changes with respect to a variable. Mathematically, if we have two functions $y=f(x)$ and $u=g(x)$, we can talk about the derivative of $y$ with respect to $u$, denoted as $\frac{dy}{du}$.

Chain Rule

The chain rule is a fundamental tool for finding the derivative of one function with respect to another. It states that if we have a composite function $h(x) = (f \circ g)(x) = f(g(x))$, then the derivative of $h$ with respect to $x$ is given by:

$$ \frac{dh}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx} $$

Using the chain rule, we can find $\frac{dy}{du}$ by first finding $\frac{dy}{dx}$ and $\frac{du}{dx}$, and then dividing the former by the latter:

$$ \frac{dy}{du} = \frac{\frac{dy}{dx}}{\frac{du}{dx}} $$

This formula assumes that $\frac{du}{dx} \neq 0$.

Implicit Differentiation

When functions are given implicitly, meaning $y$ is not explicitly solved for in terms of $x$, we can still use the concept of taking the derivative of one function with respect to another. Implicit differentiation involves differentiating both sides of an equation with respect to $x$ and then solving for $\frac{dy}{dx}$.

Table of Differences and Important Points

Aspect	Derivative w.r.t. Variable	Derivative of Function w.r.t. Another Function
Notation	$\frac{dy}{dx}$	$\frac{dy}{du}$
Definition	Rate of change of $y$ with respect to $x$	Rate of change of $y$ with respect to $u$
Chain Rule	$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$	$\frac{dy}{du} = \frac{\frac{dy}{dx}}{\frac{du}{dx}}$
Implicit Differentiation	Used when $y$ is not explicitly solved for $x$	Also applicable, but with respect to another function $u$
Application	Standard differentiation problems	Parametric equations, related rates, implicit functions

Formulas

• Chain Rule: $\frac{dh}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}$
• Derivative of $y$ with respect to $u$: $\frac{dy}{du} = \frac{\frac{dy}{dx}}{\frac{du}{dx}}$

Examples

Example 1: Parametric Equations

Consider the parametric equations $x = t^2$ and $y = t^3$. To find $\frac{dy}{dx}$, we use:

$$ \frac{dx}{dt} = 2t, \quad \frac{dy}{dt} = 3t^2 $$

Then,

$$ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{3t^2}{2t} = \frac{3}{2}t $$

Example 2: Related Rates

Suppose we have a circle with radius $r$ that is changing over time, and we want to find the rate at which the area $A$ is changing with respect to the radius. We have $A = \pi r^2$, so:

$$ \frac{dA}{dr} = 2\pi r $$

This gives us the rate of change of the area with respect to the radius.

Example 3: Implicit Differentiation

Given an equation $x^2 + y^2 = 1$, we can find $\frac{dy}{dx}$ by differentiating implicitly:

$$ 2x + 2y\frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -\frac{x}{y} $$

If we want to find $\frac{dx}{dy}$, we simply take the reciprocal:

$$ \frac{dx}{dy} = -\frac{y}{x} $$

In conclusion, the derivative of one function with respect to another function extends the concept of the derivative beyond the traditional scope of a single variable. It is a powerful tool in calculus that allows us to analyze the relationship between two changing quantities in a variety of contexts.