Section 4.5 - The Chain Rule and Implicit Differentiation

Section 4.5 - The Chain Rule and Implicit Differentiation

The Chain Rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. It allows us to find the derivative of a function that is composed of other functions, by relating the rates of change of the inner and outer functions.

• Definition: Suppose that $x = f(t)$ is a differentiable function of $t$, and $y = g(x)$ is a differentiable function of $x$. Then $y$ is a differentiable function of $t$, and the chain rule states:

$\displaystyle \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$

• Generalization: If $x$ and $y$ are both differentiable functions of $t$, then:

$\displaystyle \frac{dz}{dt} = \frac{\partial z}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial z}{\partial y} \cdot \frac{dy}{dt}$

Example 1

Let $z = x^2 y + 2xy^4$, where $x = \sin(2t)$, $y = \cos t$. Find $\frac{dz}{dt}$ when $t = 0$.

• Step 1: Compute $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.

• Step 2: Compute $\frac{dx}{dt}$ and $\frac{dy}{dt}$.

• Step 3: Substitute into the chain rule formula above.

Example 2

Find $\frac{dz}{dt}$ for $z = \sqrt{x^2 - y^2}$, $x = t^3$, $y = e^{t^4}$.

• Application: Use the chain rule for multivariable functions as above.

Multivariable Chain Rule

When dealing with functions of several variables, the chain rule can be extended. If $z = f(x, y)$, and $x$ and $y$ are both functions of $t$, then:

$\displaystyle \frac{dz}{dt} = \frac{\partial z}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial z}{\partial y} \cdot \frac{dy}{dt}$

• Variables: $t$ is the independent variable, $x$ and $y$ are called intermediate variables, and $z$ is the dependent variable.

Example 3

If $z = e^{x} \sin y$, where $x = s^2$ and $y = s^3$, find $\frac{dz}{ds}$ and $\frac{dz}{dt}$.

Example 4

Find $\frac{dz}{ds}$ and $\frac{dz}{dt}$ where $z = 3x^2 - 2xy$, $x = e^{st}$, $y = e^{t} \cos t$.

General Version of the Chain Rule

The chain rule can be generalized for functions of many variables. Suppose $z$ is a differentiable function of the $n$ variables $x_1, x_2, ..., x_n$, and each $x_j$ is a differentiable function of the $m$ variables $t_1, ..., t_m$. Then $z$ is a function of $t_1, ..., t_m$, and:

$\displaystyle \frac{\partial z}{\partial t_i} = \sum_{j=1}^n \frac{\partial z}{\partial x_j} \cdot \frac{\partial x_j}{\partial t_i}$

Implicit Differentiation

Implicit differentiation is a technique used when a function is defined implicitly by an equation involving multiple variables, rather than explicitly as $y = f(x)$.

Case 1: Two Variables

If an equation of the form $F(x, y) = 0$ defines $y$ implicitly as a differentiable function of $x$, then:

$\displaystyle \frac{dy}{dx} = -\frac{F_x}{F_y}$

where $F_x$ and $F_y$ are the partial derivatives of $F$ with respect to $x$ and $y$, respectively.

Case 2: Three Variables

If $z$ is given implicitly as a function $z = f(x, y)$ by an equation $F(x, y, z) = 0$, then:

$\displaystyle \frac{\partial z}{\partial x} = -\frac{F_x}{F_z}$

$\displaystyle \frac{\partial z}{\partial y} = -\frac{F_y}{F_z}$

where $F_x$, $F_y$, and $F_z$ are the partial derivatives of $F$ with respect to $x$, $y$, and $z$.

Example 8

Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ if $x^4 + y^4 + z^4 + e^{z} + 6xyz = 1$.

• Step 1: Compute the partial derivatives $F_x$, $F_y$, and $F_z$.

• Step 2: Substitute into the formulas above to find the required derivatives.

Summary Table: Chain Rule and Implicit Differentiation

Technique	Formula	Application
Chain Rule (Single Variable)	$\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$	Differentiating composite functions
Chain Rule (Multivariable)	$\frac{dz}{dt} = \frac{\partial z}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial z}{\partial y} \cdot \frac{dy}{dt}$	Functions of several variables
Implicit Differentiation (2 variables)	$\frac{dy}{dx} = -\frac{F_x}{F_y}$	Implicitly defined functions
Implicit Differentiation (3 variables)	$\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}$ $\frac{\partial z}{\partial y} = -\frac{F_y}{F_z}$	Implicitly defined functions of three variables

Additional info: The notes cover both the standard and multivariable chain rule, as well as implicit differentiation for two and three variables, with worked examples and summary formulas. These topics are essential for understanding differentiation in calculus, especially for functions defined implicitly or as compositions.