What is the Chain Rule?

The Chain Rule is a mathematical method to differentiate a composition of functions. From this composition of functions, we can discern the functions’ derivatives and their relationships. 

Composition of Functions

A composition of functions is when one variable depends on another variable, which itself depends on another variable. Below are different ways to express this composition of functions. 

• If y = f(x) and z = g(y), then z depends on y, and y depends on x.
• This relationship can also be represented in fractional form.
  • • $\dfrac{\text{change in } z}{\text{change in } x}$ = $\dfrac{\text{change in } z}{\text{change in } y}$ $\times$ $\dfrac{\text{change in } y}{\text{change in } x}$
• In addition, Leibniz Notation more succinctly describes this equation.
  • • Leibniz Notation: $\dfrac{dz}{dx}$ = $\dfrac{dz}{dy}$ $\times$ $\dfrac{dy}{dx}$

The Chain Rule Formula

• The Chain Rule formula is $\dfrac{d}{dx}$ [f(g(x))] = f'(g(x)) $\times$ g'(x)
  • • In other words, the derivative of the composite function = derivative of the outside function $\times$ derivative of the inside function

Practice with the Chain Rule Formula

The Chain Rule can be used to differentiate many types of functions. Here I have listed 4 practice problems. In all 4 problems, we use the same basic formula, although we find the individual derivatives in different ways.

• 2 Basic Functions – Use the Chain Rule to differentiate ($x^{3} +2)^{5}$
  • • First, assign a term to the outside f(x) and inside g(x) functions
      • • KEY: a helpful step to decide which function is which is boxing the inner function
        • outside function f(x) = $x^{5}$
        • inside function g(x) = $x^{3}+2$, with a box around it
    • Plug in the f(x) and g(x) into the Chain Rule formula
      • • $\frac{d}{dx}$[f(g(x))] = f'(g(x)) $\times$ g'(x)
        • $5(x^{3} +2)^{4}$ $\times$ $3x^{2}$

• The Number e – Use the Chain Rule to differentiate y = $e^{x^3+1}$
  • • Assign terms
      • • outside function f(x) = $e^{x}$
        • inside function g(x) = $x^{3}+1$, with a box around it
    • Plug into formula
      • • $\frac{d}{dx}$[f(g(x))] = f'(g(x)) $\times$ g'(x)
        • $e^{x^3+1}$ $\times$ $3x^{2}$

• Trig Functions – Use the Chain Rule to differentiate f(x) = $e^{6\sin(4x)}$
  • • Assign terms
      • • outside function f(x) = $e^{x}$
        • inside function g(x) = ${6\sin(4x)}$, with a box around it
    • Plug into formula
      • • $\frac{d}{dx}$[f(g(x))] = f'(g(x)) $\times$ g'(x)
        • KEY: the derivative of ${\sin(x)}$ =  ${\cos(x)}$, ${\cos(x)}$ = ${-\sin(x)}$
        • $e^{6\sin(4x)}$ $\times$ 24${\cos(4x)}$

• Natural Log (ln) – Use the Chain Rule to differentiate $ln(3p^{2}-5)$
  • • Assign terms
      • • outside function f(x) = ln(x)
        • inside function g(x) = $3p^{2}-5$, with a box around it
    • Plug into formula
      • • $\frac{d}{dx}$[f(g(x))] = f'(g(x)) $\times$ g'(x)
        • KEY: the derivative of ln(x) = 1/x
        • $\frac{1}{3p^{2}-5}$ $\times$ 6p

Evaluating Graphs with the Chain Rule

Evaluating graphs involves a different set of skills. The derivative of a function is the slope of the tangent line. In these problems, we must locate the point, and then measure the slope of its tangent line. From there, we can apply the Chain Rule.

Let f and g be functions. Let y=f(x) and z=g(y)

y=f(x) function

z=g(y) function

\

 Find $\frac{dz}{dx}$ when x = 1

In order to find this answer, we will be using the equation $\dfrac{dz}{dx}$ = $\dfrac{dz}{dy}$ $\times$ $\dfrac{dy}{dx}$

• • locate the y value when x=1 on y=f(x) function
    • • here it would be y=2
      • y=2 will be helpful for locating the other function’s coordinate
  • find the slope of $\dfrac{dy}{dx}$
    • • m = -2
  • using y=2, find the y value at x=2 on z=g(y) function
    • • y=2 again
  • find the slope of $\dfrac{dz}{dy}$
    • • m = 1
  • multiply these slopes together
    • • $\dfrac{dz}{dx}$ =  $\dfrac{dz}{dy}$ $\times$ $\dfrac{dy}{dx}$
      • -2 $\times$ 1 = -2

Evaluating Functions with Their Derivatives

This section is easier than the previous graphing section. We are given the functions and their derivatives. Now, we just plug in!

Let h(x) = f(g(x)) and k(x) = g(f(x))

f(2)=3    f'(0)=0    f'(2)=1   g(2)=0    g'(2) = 3    g'(3) = -2

Evaluate h'(2)

• Chain Rule Formula: $\dfrac{d}{dx}$ [f(g(x))] = f'(g(x)) $\times$ g'(x)
• Equivalently, h'(x) = f'(g(x)) $\times$ g'(x) here
  • • h'(2) = f'(g(2)) $\times$ g'(2)
    • h'(2) = f'(0) $\times$ g'(2)
    • h'(2) = (0) $\times$ (3)
    • h'(2) = 0

Evaluate k'(2)

• Chain Rule Formula: $\dfrac{d}{dx}$ [f(g(x))] = f'(g(x)) $\times$ g'(x)
• Equivalently, k'(x) = g‘(f(x)) $\times$ f'(x) here
  • • k'(2) = g'(f(2)) $\times$ f'(2)
    • k'(2) = g'(3) $\times$ f'(2)
    • k'(2) = (-2) $\times$ (1)
    • k'(2) = -2

Real World Applications of the Chain Rule

The Chain Rule can also help us deduce rates of change in the real world. From the Chain Rule, we can see how variables like time, speed, distance, volume, and weight are interrelated.

A horse is carrying a carriage on a dirt path. The amount of ${\text(energy)}$ E (in calories) expended by the horse depends on the ${\text(distance)}$ m (in miles) the horse walks. Also, the distance walked by the horse depends on ${\text(time)}$ t (in hours). If the horse expends 40 calories per mile and the horse walks at a speed of 8 mph, at what rate is the horse expending energy?

• Chain Rule Formula: $\dfrac{dz}{dx}$ = $\dfrac{dz}{dy}$ $\times$ $\dfrac{dy}{dx}$
• —————————————————————————————–
• $\dfrac{dE}{dm}$ = $\dfrac{energy}{time}$
• —————————————————————————————–
• $\dfrac{dE}{dm}$ $\times$ $\dfrac{dm}{dt}$
• —————————————————————————————–
• $\dfrac{40 calories}{1 mile}$ $\times$ $\dfrac{8 miles}{1 hour}$ = 500 calories
• —————————————————————————————–
• The horse expends energy at a rate 500 calories per 1 hour of walking.