# Computing Derivatives Examples

#### Lines

We've seen lines before. The restroom at half-time, the blue things on notebook paper...and they were all over algebra. y = mx + b, anyone? Lines are a special case of polynomial...

#### Power Functions

A power function is any function of the form f(x) = xa, where a is any real number. Sample ProblemThe following are all power functions:Sample ProblemThe following are all power functio...

#### Logarithmic Functions

Logarithmic functions come in different flavors and bases. There's vanilla, strawberry...sugar cone, dish...but we'll start with the most basic one: f(x) = ln x.This function is defined for x > 0...

#### Derivative of a Constant Multiple of a Function

We'll start with taking the derivatives of some lines.Sample ProblemFind the derivative of each function.f(x) = x g(x) = 3f(x) h(x) = 3g(x)Answer. These are all lines.The de...

#### Fractions With a Constant Denominator

If we have a fraction where c is a constant, this is the same thing as . When asked to take derivatives of functions like this, the first thing we do is rewrite the original funct...

#### More Derivatives of Logarithms

Whenever possible, rewrite a function so it looks good before taking its derivative.Any logarithm logb a can be written asThis allows us to find derivatives of logarithms in bases besides e....

#### Derivative of a Sum (or Difference) of Functions

Anytime we add two functions, we can find the derivative of the sum pretty easily, so long as we know the derivative of each function.The derivative of a sum is just the sum of the derivatives:
(...

#### Derivative of a Product of Functions

There's a convenient rule we can use to find the derivative of expressions like (3x)(sin x) or x2ex.For the first expression, we know how to find the derivatives of 3x and sin x, but we don't yet...

#### Derivative of a Quotient of Functions

Division within derivatives is more complicated than the other rules we've seen so far. Make some space in the ol' memory bank for the Quotient Rule. The Quotient Rule states that the deriva...

#### Using the Correct Rule(s)

"Work from the outside in" is a suggestion for how to organize our work when the derivatives become more difficult to manage. We want to start with the outer-most operation, and work in from ther...

#### The Chain Rule

When dealing with a compositionh(x) = f(g(x)),the function f is called the outside function and the function g is called the inside function. f is the outside function because it's written on the...

#### Derivatives of Inverse Trigonometric Functions

Finding derivatives of inverse trig functions will be similar to finding the derivative of ln x. Our main tool is the chain rule. We also need some background information:the fact that composing in...

#### The Chain Rule in Leibniz Notation

We stated the chain rule first in Lagrange notation. Since Leibniz notation lets us be a little more precise about what we're differentiating and what we're differentiating with respect to, we ne...

#### Patterns

Combining the derivatives of basic functions with the chain rule gives us a lot of patterns that let us take derivatives of functions that seem complicated. Sample Problem Let h(x) = ecos x. If w...

#### Using Leibniz Notation

Some things to remember for implicit differentiation:Since y is a function of x, any derivative involving y must use the chain rule. Since y is a function of x, taking the derivative o...

#### Using Lagrange Notation

Not a fan of Leibniz notation? We can do implicit differentiation with Lagrange notation just as well.Things to remember for implicit differentiation with Lagrange notation:x' = 1.since y is a fu...