MTH1150 Rules of Differentiation

Rules of Differentiation So far we have learned that in order to find a function's derivative we need to use The Newton Quotient. This is true, but in practice it is often very difficult to pass higher order functions through The Newton Quotient. There must be an easier way?

The Differentiation Rules Using The Newton Quotient the rules of differentiation have been derived. Using these rules you will be able to differentiate almost any function, regardless of the function’s complexity.

The Rules of Differentiation: General Power Rule 𝑑 𝑑𝑥 𝑥 𝑟 =𝑟 𝑥 𝑟−1   Reciprocal Rule: 𝑑 𝑑𝑥 1 𝑓(𝑥) =− 𝑓′(𝑥) 𝑓(𝑥) 2 Summation: 𝑑 𝑑𝑥 𝑓 𝑥 ±𝑔(𝑥) = 𝑓 ′ 𝑥 ±𝑔′(𝑥) Quotient Rule: 𝑑 𝑑𝑥 𝑓(𝑥) 𝑔(𝑥) = 𝑔 𝑥 ∙ 𝑓 ′ 𝑥 −𝑓(𝑥)∙𝑔′(𝑥) 𝑔(𝑥) 2 Constant Multiples: 𝑑 𝑑𝑥 𝑐∙𝑓(𝑥) =𝑐∙𝑓′(𝑥) Chain Rule: 𝑑 𝑑𝑥 𝑓 𝑔(𝑥) =𝑓′ 𝑔 𝑥 ∙𝑔′(𝑥) Product Rule: 𝑑 𝑑𝑥 𝑓(𝑥)∙𝑔(𝑥) = 𝑓 ′ 𝑥 ∙𝑔 𝑥 +𝑓(𝑥)∙𝑔′(𝑥)

Using the Rules Every function that you come across will fit into at least one of the differentiation rules. The problem now lies with deciding which rule, or combination of rules to use.

The General Power Rule The general power rule is the first and most used of the differentiation rules. Using the general power rule we can differentiate any variable to the power of r. General Power Rule 𝑑 𝑑𝑥 𝑥 𝑟 =𝑟 𝑥 𝑟−1

Example 1 Find the derivatives of the following functions: 𝑓 𝑥 = 𝑥 3 𝑓 𝑥 = 𝑥 3 𝑦= 𝑥 7 𝑓 𝑥 = 𝑥 100

The Summation Rule The summation rule says that the derivative of the sum of two (or more) functions is the sum of the two (or more) functions' derivatives. Summation: 𝑑 𝑑𝑥 𝑓 𝑥 ±𝑔(𝑥) = 𝑓 ′ 𝑥 ±𝑔′(𝑥)

Example 2 Find the derivative of the following functions: 𝑓 𝑥 = 𝑥 3 + 𝑥 5 + 𝑥 7 𝑓 𝑥 = 𝑥 100 + 𝑥 200 + 𝑥 300

Constant Multiples The rule of constant multiples says that the derivative of a function multiplied by a constant is that function’s derivative multiplied by the constant. Constant Multiples: 𝑑 𝑑𝑥 𝑐∙𝑓(𝑥) =𝑐∙𝑓′(𝑥)

Example 3 Find the derivatives of the following functions: 𝑓 𝑥 = 3𝑥 2 + 2𝑥+ 1 𝑓 𝑥 = 100𝑥 4 + 200 𝑥 5 + 1000

The Product Rule The product rule allows us to find the derivative of two functions multiplied together. Note that it isn’t just as simple as multiplying the two derivatives together. Product Rule: 𝑑 𝑑𝑥 𝑓(𝑥)∙𝑔(𝑥) = 𝑓 ′ 𝑥 ∙𝑔 𝑥 +𝑓(𝑥)∙𝑔′(𝑥)

The Product Rule Once it has been determined that the product rule is required, follow these steps to compute the derivative: Step 1: Break the equation into its two functions. Let one function be called f(x), and the other g(x). Step 2: Find the derivatives of f(x) and g(x). Step 3: Plug these functions into The Product Rule. Step 4: Simplify the equation.

Example 4 Find the derivative of the following function: 𝑦=( 𝑥 2 +1)( 𝑥 3 −1)

The Reciprocal Rule The reciprocal rule allows us to find the derivative of the reciprocal of a function. Note that it isn’t as simple as taking the inverse of the derivative. Reciprocal Rule: 𝑑 𝑑𝑥 1 𝑓(𝑥) = −𝑓′(𝑥) 𝑓(𝑥) 2

The Reciprocal Rule Once it has been determined that the reciprocal rule is required, follow these steps to compute the derivative. Step 1: Let the denominator of the function be called f(x). Step 2: Find the derivative of f(x). Step 3: Plug the sub functions into The Reciprocal Rule. Step 4: Simplify the equation.

Example 5 Find the derivative of the following function: 𝑦= 1 3 𝑥 2 −2𝑥+4

The Quotient Rule The quotient rule allows us to find the derivative of an equation defined by the ratio of two functions of x. Note that it isn’t as simple as finding the derivative of the numerator and denominator and then dividing them. Quotient Rule: 𝑑 𝑑𝑥 𝑓(𝑥) 𝑔(𝑥) = 𝑔 𝑥 ∙ 𝑓 ′ 𝑥 −𝑓(𝑥)∙𝑔′(𝑥) 𝑔(𝑥) 2

The Quotient Rule Once it has been determined that the quotient rule is required, follow these steps to compute the derivative. Step 1: Break the equation into its two functions. Let the numerator be called f(x), and the denominator g(x). Step 2: Find the derivatives of f(x) and g(x). Step 3: Plug these functions into The Quotient Rule. Step 4: Simplify the equation.

Example 6 Find the derivative of the following function: 𝑦= 1− 𝑥 2 1+ 𝑥 2

Multiple Rules Often times you will be faced with finding the derivative of a function that requires more than one of the rules of differentiation. The key is to first determine which of the rules you need, and then in which order to apply them. You need to find the primary form of the function.

Multiple Rules (General Approach) Start with the quotient rule. If you are dealing with a rational function you will need to apply the quotient rule before any other. As you determine the derivative of the numerator and the denominator you may need to use the product rule or the chain rule.

Multiple Rules The best practice for determining the order in which to apply the differentiation rules is to decide on the order of operations (BEDMAS) that would be required to evaluate the function at a point. The last operation that you would preform is the first differentiation rule that you need to apply.

Example 7 Find the derivative of the following function: 𝑦= (1− 𝑥 2 )∙𝑥 1+ 𝑥 2

Derivatives of Trig Functions