Notation & Definition of the Derivative

Understanding the Derivative

• The operator means “differentiate (something) with respect to .” It is meaningless on its own.

• Given a function , the derivative of this function can be written in three ways:

• is the slope of a tangent line to the graph of at a particular , or the instantaneous rate of change of with respect to .

Standard Derivatives

Common Derivative Formulas

• ,

Derivatives of Sums and Scalar Multiples

Linearity of the Derivative

• , where

Product Rule

Derivative of a Product

• If and are differentiable functions, then:

Quotient Rule

Derivative of a Quotient

• If and are differentiable functions, then:

Chain Rule

Derivative of a Composite Function

• If can be written as a function of (the ‘inner function’), where $u$ is a function of , then:

• In general, we do not need to use labels such as . Instead, we think in terms of ‘inner’ and ‘outer’ functions.

• Hence,

The derivative of a composite function is equal to the derivative of the outer function (evaluated at the inner function) times the derivative of the inner function.

Implicit Differentiation

Differentiating Implicit Relations

• Used to find derivatives of general relations of the form that cannot be rearranged into the form .

• We say that implicitly defines to be a function of .

Procedure: Given the implicit equation :

1. Differentiated both sides of the equation with respect to (without thinking of as a function of $x$). Expressions involving $y$ require the use of the chain rule.

2. Collect the terms in on one side of the equation.

3. Factor out .

4. Divide both sides by the coefficient of .

Logarithmic Differentiation

Using Logarithms to Simplify Differentiation

• Take the logarithm of a complex expression (involving products, quotients, and exponents) to differentiate it more easily (thereby splitting it into more manageable parts) and then differentiate implicitly.

• The general power function, , cannot be differentiated using the Power Rule for derivatives. Instead, use logarithmic differentiation.

• First take natural logs of both sides: , then differentiate both sides.

Derivatives of Hyperbolic Functions

Standard Hyperbolic Derivatives

Derivatives of Inverse Hyperbolic Functions

Standard Inverse Hyperbolic Derivatives

Note: All derivatives are given on the 6-page MATH1020 Calculus for Engineers Formulae sheet.