2.8 Implicit Differentiation Definition. We will say that a given equation in x and y defines the function f implicitly if the graph of y = f(x) coincides with a portion of the graph of the equation. Example: The equation implicitly defines functions The equation implicitly defines the functions

Two differentiable methods There are two methods to differentiate the functions defined implicitly by the equation. For example: One way is to rewrite this equation as , from which it follows that

Two differentiable methods The other method is to differentiate both sides of the equation before solving for y in terms of x, treating y as a differentiable function of x. The method is called implicit differentiation. With this approach we obtain Since ,

Implicit Differentiation

Example Example: Use implicit differentiation to find dy / dx if Solution:

Example Example: Find dy / dx if Solution:

2.10 Logarithmic Functions

Logarithm Function with Base a

Natural Logarithm Function Logarithms with base e and base 10 are so important in applications that Calculators have special keys for them. logex is written as lnx log10x is written as logx The function y=lnx is called the natural logarithm function, and y=logx is Often called the common logarithm function.

Properties of Logarithms

Properties of ax and logax

Derivative of the Natural Logarithm Function Note: Example: Solution:

Example Example: Solution:

Derivatives of au Note that Example:

Derivatives of logau Note that Example:

The Number e as a Limit

2.11 Inverse Trigonometric Functions The six basic trigonometric functions are not one-to-one (their values Repeat periodically). However, we can restrict their domains to intervals on which they are one-to-one.

Six Inverse Trigonometric Functions Since the restricted functions are now one-to-one, they have inverse, which we denoted by These equations are read “y equals the arcsine of x” or y equals arcsin x” and so on. Caution: The -1 in the expressions for the inverse means “inverse.” It does Not mean reciprocal. The reciprocal of sinx is (sinx)-1=1/sinx=cscx.

Derivative of y = sin-1x Example: Find dy/dx if Solution:

Derivative of y = tan-1x Example: Find dy/dx if Solution:

Derivative of y = sec-1x Example: Find dy/dx if Solution:

Derivative of the other Three There is a much easier way to find the other three inverse trigonometric Functions-arccosine, arccotantent, and arccosecant, due to the following Identities: It follows easily that the derivatives of the inverse cofunctions are the negatives of the derivatives of the corresponding inverse functions.