Quotient & Chain Rules

What are Quotient & Chain Rules?

The quotient rule is a method in calculus for determining the derivative or differentiation of a function given as a ratio or division of two differentiable functions. That is, we can use the quotient rule to find the derivative of a function of the form: f(x)/g(x), provided that both f(x) and g(x) are differentiable and g(x) equals 0. The quotient rule directly follows the product rule as well as the concept of boundaries of derivation in differentiation. In the following sections, we will go over the formula for the quotient rule, as well as its proof using solved examples.

Definition of Quotient Rule

In calculus, the quotient rule is a method for calculating the derivative of any function given in the form of a quotient obtained from the division of two differentiable functions. According to the quotient rule, the derivative of a quotient is equal to the ratio of the result obtained by subtracting the numerator times the denominator’s derivative from the denominator times the denominator’s derivative to the square of the denominator.

Quotient Rule Formula Derivation

We learned about the quotient formula in the previous section, which is used to find derivatives of the quotient of two differentiable functions. Let us now look at the proof of the quotient rule formula. There are several methods for proving the quotient rule formula, which is as follows:

  1. Making use of derivative and limit properties
  2. Making use of implicit differentiation
  3. Applying the chain rule

What is Chain Rule?

The chain rule provides a method for determining the derivative of composite functions, with the number of functions in the composition determining how many differentiation steps are required.

How Does Chain Rule Work?

This chain rule is also referred to as the outside-in rule, the composite function rule, or the function of a function rule. It is only used to compute the derivatives of the composite functions. To understand the concept of Quotient Rules and Chain Rules, you can also visit the Cuemath website.

The Chain Rule Theorem: Assume f is a real-valued function that is a composite of two functions g and h. That is, f = g o h. If u = h(x) and du/dx and dg/du exist, then this could be written as:

change in f/change in x = change in g/change in u/change in x

This is expressed in Leibniz notation as an equation, df/dx = dg/du.du/dx.

Chain Rule Steps

  • Step 1: Determine The Chain Rule: The function must be a composite function, which means it is nested over another.
  • Step 2: Determine the inner and outer functions.
  • Step 3: Leaving the inner function alone, find the derivative of the outer function.
  • Step 4: Determine the inner function’s derivative.
  • Step 5: Add the results from steps 4 and 5.
  • Step 6: Reduce the complexity of the chain rule derivative.

As an example: Consider the following function: g(x) = ln (sin x)

  • The function g is a composite function. Use the chain rule as a result.
  • Sin x is the inner function, and ln is the outer function (x).
  • The outer function’s derivative is 1/sin x.
  • The inner function’s derivative is cos x.
  • Finally, g'(x) = the derivative of the outside function while leaving the inside function alone; the derivative of the inside function = 1/sin x cos x
  • When we simplify, we get cos x/sin x = cot x.

Applications of The Chain Rule

This chain rule has numerous applications in physics, chemistry, and engineering. We use the chain rule:

  • To calculate the pressure’s time rate of change,
  • In order to compute the rate of change of distance between two moving objects,
  • To determine the position of an object that moves to the right and left in a specific interval,
  • To determine if a function is increasing or decreasing, we use this formula
  • To calculate the average molecular speed’s rate of change,

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