The Chain Rule - AP Calculus BC Study Guide

The Core Idea: The Chain Rule

The chain rule is a fundamental tool for finding the derivative of a composite function—that is, a function that is formed by the composition of two or more functions, often described as a "function within a function." When a variable $x$ is acted upon by an "inner" function, and the result of that is then acted upon by an "outer" function, the chain rule provides a method to determine the instantaneous rate of change of the overall composite function.

This process involves differentiating both the outer and inner functions in a specific sequence. The rule states that the derivative of the composite function is the product of the derivative of the outer function (evaluated at the inner function) and the derivative of the inner function. This principle is foundational and can be applied repeatedly for functions with multiple layers of composition.

Key Formulas

The chain rule can be expressed in two common and equivalent notations.

1. Function Notation:

   If a function $h(x)$ is a composition of two differentiable functions $f$ and $g$ such that $h(x)=f(g(x))$, its derivative is given by:

   $$h^{\prime }(x)=\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$$

   This formula is often described as "the derivative of the outside function, evaluated at the inside function, times the derivative of the inside function."

2. Leibniz Notation:

   If $y$ is a function of $u$, say $y=f(u)$, and $u$ is a function of $x$, say $u=g(x)$, then $y$ is a composite function of $x$. The derivative of $y$ with respect to $x$ is given by:

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

   This notation highlights how the "links" in the chain ($du$) connect the rate of change of $y$ with respect to $u$ to the rate of change of $u$ with respect to $x$.

Understanding Applications of the Rule

The true power of the chain rule lies in its versatility. The functions involved in the composition, $f(x)$ and $g(x)$, do not need to be defined by symbolic formulas. The chain rule is a structural rule that applies regardless of how the functions are presented.

• Functions Defined by Tables: You may be given a table of values for $f(x)$, $g(x)$, $f^{\prime }(x)$, and $g^{\prime }(x)$ at several points. To find the derivative of a composition like $h(x)=f(g(x))$ at a specific point $x=a$, you must use the table to look up the necessary values for $g(a)$, $g^{\prime }(a)$, and $f^{\prime }(g(a))$.

• Functions Defined by Graphs: Similarly, if $f(x)$ and $g(x)$ are presented as graphs, you can find the derivative of their composition. This involves reading function values (y-coordinates) and derivative values (slopes of tangent lines at specific points) directly from the graphs to use in the chain rule formula.

• Multiple Applications: The chain rule is not limited to a composition of two functions. If a function is a composition of three or more functions, such as $h(x)=f(g(k(x)))$, the chain rule is applied iteratively from the outermost function to the innermost function:

  $$h^{\prime }(x)=f^{\prime }(g(k(x)))\cdot g^{\prime }(k(x))\cdot k^{\prime }(x)$$

  Each "layer" of the function is differentiated and multiplied in sequence.

Core Concepts & Rules

• Purpose: The chain rule is the required method for differentiating composite functions of the form $f(g(x))$.

• The Rule: The derivative of a composite function is the product of two rates: the derivative of the outer function evaluated at the inner function, and the derivative of the inner function.

• Versatility: The rule applies whether the functions are defined symbolically (e.g., $f(x)=x^2$), by a table of values, or by a graph.

• Nesting: For compositions of more than two functions, such as $f(g(h(x)))$, the chain rule is applied sequentially. You differentiate the outermost function, then multiply by the derivative of the next function in, and so on, until you have multiplied by the derivative of the innermost function.

Step-by-Step Example 1: Basic Application

Problem: Find the derivative of the function $h(x)=\cos (x^4)$.

Step 1: Identify the outer and inner functions.

The function $h(x)$ is a composition $f(g(x))$.

• The outer function is what is done last: $f(u)=\cos (u)$.

• The inner function is what is done first: $g(x)=x^4$.

Step 2: Find the derivatives of the outer and inner functions separately.

• Derivative of the outer function: $f^{\prime }(u)=-\sin (u)$.

• Derivative of the inner function: $g^{\prime }(x)=4x^3$.

Step 3: Apply the chain rule formula: $h^{\prime }(x)=f^{\prime }(g(x))\cdot g^{\prime }(x)$.

• First, find $f^{\prime }(g(x))$. This means we take the derivative of the outer function, $f^{\prime }(u)=-\sin (u)$, and substitute the original inner function $g(x)=x^4$ back in for $u$.

  $f^{\prime }(g(x))=-\sin (x^4)$.

• Now, multiply by the derivative of the inner function, $g^{\prime }(x)=4x^3$.

Step 4: Combine the pieces and simplify.

$$h^{\prime }(x)=f^{\prime }(g(x))\cdot g^{\prime }(x)=-\sin (x^4)\cdot (4x^3)$$

$$h^{\prime }(x)=-4x^3\sin (x^4)$$

Step-by-Step Example 2: Exam-Style Application

Problem: The table below gives values for differentiable functions $f$ and $g$, and their derivatives, at selected values of $x$.

If $h(x)=f(g(x))$, find $h^{\prime }(1)$.

Step 1: Write down the general derivative using the chain rule.

The function is $h(x)=f(g(x))$. Applying the chain rule gives:

$$h^{\prime }(x)=f^{\prime }(g(x))\cdot g^{\prime }(x)$$

Step 2: Substitute the specific value of $x$ into the derivative formula.

We need to find $h^{\prime }(1)$, so we substitute $x=1$:

$$h^{\prime }(1)=f^{\prime }(g(1))\cdot g^{\prime }(1)$$

Step 3: Evaluate the inner parts of the expression using the table.

First, we need to find the value of the inner function, $g(1)$. Looking at the table in the row for $x=1$, we see that $g(1)=2$.

We also need the value of $g^{\prime }(1)$. From the same row, we see that $g^{\prime }(1)=3$.

Step 4: Substitute the value of $g(1)$ back into the expression.

Our expression becomes:

$$h^{\prime }(1)=f^{\prime }(2)\cdot g^{\prime }(1)$$

Step 5: Find the remaining values from the table.

Now we need to find $f^{\prime }(2)$. Looking at the table in the row for $x=2$, we find that $f^{\prime }(2)=-4$. We already found $g^{\prime }(1)=3$.

Step 6: Calculate the final result.

Substitute the final values into the expression:

$$h^{\prime }(1)=(-4)\cdot (3)=-12$$

Using Your Calculator

The chain rule is an analytical process for finding a derivative function or its value. A calculator cannot perform the symbolic steps of applying the chain rule. However, it is an excellent tool for checking your answer after you have found the derivative by hand.

To check the result of a derivative at a single point, you can use the numerical derivative feature.

Example Check: Let's verify the result from Example 1, $h^{\prime }(x)=-4x^3\sin (x^4)$, at $x=1$.

1. By Hand:$h^{\prime }(1)=-4(1{)}^3\sin (1^4)=-4\sin (1)\approx -3.366$.

2. On the Calculator (TI-84 style):

   • Press MATH and select nDeriv( (or $8$).

   • Enter the expression, variable, and point of evaluation: nDeriv(cos(X^4), X, 1)

   • The calculator will return a value approximately equal to $-3.366$.

Since the results match, you can be confident in your analytical answer. You can also graph both your derived function and the numerical derivative of the original function to see if the graphs overlap, which would confirm your derivative is correct for all values of $x$.

AP Exam Quick Hit

Common Question Types

• Derivatives from a Table: You will be given a table of function and derivative values for $f$ and $g$ and asked to find the derivative of a composite function like $h(x)=f(g(x))$ or $k(x)=(g(x){)}^3$ at a specific point. This tests your ability to apply the rule and correctly retrieve values from the table.

• Derivatives from a Graph: You will be shown graphs of functions $f$ and $g$ (often piecewise linear) and asked to find the derivative of their composition, $h(x)=f(g(x))$, at a point $x=a$. This requires you to find $g(a)$ (the y-value on the graph of g) and $g^{\prime }(a)$ and $f^{\prime }(g(a))$ (the slopes of the tangent lines at the relevant points).

• Nested Chain Rule with Symbolic Functions: You will be asked to find the derivative of a function with multiple "layers," requiring repeated application of the chain rule. For example, finding the derivative of $h(x)=\sqrt{\tan (e^{2x})}$.

Common Mistakes

• Forgetting to Multiply by the Inner Derivative: The most common mistake is to only differentiate the "outside" function. For example, incorrectly stating that the derivative of $e^{x^2}$ is $e^{x^2}$ instead of the correct $e^{x^2}\cdot 2x$.

• Evaluating the Outer Derivative at the Wrong Point: A frequent error is to write the chain rule as $f^{\prime }(x)\cdot g^{\prime }(x)$ instead of the correct $f^{\prime }(g(x))\cdot g^{\prime }(x)$. The derivative of the outer function must be evaluated at the original inner function, not at $x$.

• Table Value Confusion: When asked to find $h^{\prime }(a)=f^{\prime }(g(a))\cdot g^{\prime }(a)$ using a table, students often mistakenly calculate $f^{\prime }(a)\cdot g^{\prime }(a)$. You must first find the value of $g(a)$ and use that result as the input for the function $f^{\prime }$.

• Misidentifying Layers: In a complex, nested function like ${\sin }^3(4x)$, which is $(\sin (4x){)}^3$, students may fail to see all three layers: the cubing function (outermost), the sine function (middle), and the linear function $4x$ (innermost). Missing a layer results in an incorrect derivative.