- Topics At a Glance
**Derivatives of Basic Functions**- Constant Functions
- Lines
- Power Functions
- Exponential Functions
- Logarithmic Functions
**Trigonometric Functions**- Derivatives of More Complicated Functions
- Derivative of a Constant Multiple of a Function
- Multiplication by -1
- Fractions With a Constant Denominator
- More Derivatives of Logarithms
- Derivative of a Sum (or Difference) of Functions
- Derivative of a Product of Functions
- Derivative of a Quotient of Functions
- Derivatives of Those Other Trig Functions
- Solving Derivatives
- Using the Correct Rule(s)
- Giving the Correct Answers
- Derivatives of Even More Complicated Functions
- The Chain Rule
- Re-Constructing the Quotient Rule
- Derivative of a
^{x} - Derivative of
*ln*x - Derivatives of Inverse Trigonometric Functions
- The Chain Rule in Leibniz Notation
- Patterns
- Thinking Backwards
- Implicit Differentiation
- Computing Derivatives Using Implicit Differentiation
- Using Leibniz Notation
- Using Lagrange Notation
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

The **trigonometric functions** are the functions used often in trigonometry:

sin(*x*), cos(*x*), tan(*x*), csc(*x*), sec(*x*), and cot(*x*).

We'll find the derivatives of some of these now, and some we'll leave until we've learned more rules.

Start with *f*(*x*) = sin(*x*). While we *could* calculate the derivative of this function using the limit definition, we'll do what we did with the logarithmic function and use pictures.

The graph of *f*(*x*) = sin(*x*) looks like this:

Wherever the function hits a maximum or minimum value, it has a horizontal tangent line:

Wherever there's a horizontal tangent line, the derivative is zero:

Estimating *f*'(0) = 1, *f'*(π) = -1, and *f'*(2π) = 1 (Exercise: estimate these derivatives using tables.) we find the following rough graph of the derivative:

Since we already know the answer, we'll connect the dots.:

And ... da da da dum... the derivative of *f*(*x*) = sin(*x*) is

*f'*(*x*) = cos(*x*).

Now look at *f*(*x*) = cos(*x*):

Again, we can find all the places the derivative is zero by finding all the maxima and minima of *f*:

Estimating with tables we find *f'*(0) = 0, *f'*(π/2) = -1, and *f'*(3π/2) = -1 (Exercise: estimate these derivatives using tables):

Filling in the dots, we find

The derivative of *f*(*x*) = cos(*x*) is

*f'*(*x*) = -sin(*x*).

**Be Careful:** To remember that the derivative of sin(*x*) is +cos(*x*), and the derivative of cos(*x*) is -sin(*x*), look at the graphs. The graph of sin(*x*) starts by increasing, its derivative must be positive at first:

The graph of cos(*x*) starts by decreasing, its derivative must start out negative:

The other trig functions can all be written as quotients involving the sin and/or cos functions:

To find the derivatives of these functions, first we need to know how to find the derivatives of quotients.

Deriving inverse trig functions, such as arcsin(*x*) and arccos(*x*), requires knowledge of derivatives of inverse functions in general. This uses the chain rule, which we'll discuss later.

Exercise 1

What is the largest the slope of *f*(*x*) = sin(*x*) ever gets? The smallest?

Exercise 2

What is the largest the slope of *f*(*x*) = cos(*x*) ever gets? The smallest?

Exercise 3

Let *f*(*x*) = -sin(*x*). Based on the discussion in this section, what is a reasonable formula for *f'*(*x*)?

Exercise 4

Let *f*(*x*) = -cos(*x*). Based on the discussion in this section, is a reasonable formula for *f'*(*x*)?

Exercise 5

Find the derivative of the function. Be sure to use correct notation.

*f*(*x*) = sin(*x*)

Exercise 6

Find the derivative of the function. Be sure to use correct notation.

*g*(*z*) = cos(*z*)

Exercise 7

Find the derivative of the function. Be sure to use correct notation.

*h*(*y*) = -sin(*y*)

Exercise 8

Find the derivative of the function. Be sure to use correct notation.

*r*(k) = -cos(*k*)