Derivatives

1. Algebraic Deffintion

1.1. Finding the derivative by the limit process

1.2. f'x)=lim as h-->0 (f(x+h)-f(x)/h

1.3. Using the derivative to find the slope at a point

1.4. First find f'(x) and then use the given point to solve for slope.

2. Graphical Deffiniton

2.1. From viewing a graph you can determine a derivative.

2.2. Sharp turns indicate a derivative does not exist at that point.

3. Notation

3.1. f'(x)= lim as h-->0 f(x+h)-f(x)/h

3.2. f'(x)

3.3. dy/dx

3.4. d/dx[f(x)]

4. Special Rules

4.1. (d/dx)[cosx]=-sinx

4.2. (d/dx)[sinx]=cosx

4.3. (d/dx)[e^x]=e^x

5. Rules

5.1. The derivative only exist provided that the limit exist.

5.2. For all x values for which the limit exist f' is a function of x.

5.3. Constant Rule

5.3.1. (d/dx) [c]=0

5.4. Power Rule

5.4.1. (d/dx)[x^n] =nx^n-1

5.5. Constant Multiple Rule

5.5.1. (d/dx)[cf(x)]=cf'(x)

5.6. Sum and Difference Rule

5.6.1. (d/dx)[f(x)+or-g(x)]=f'(x)+or-g'(x)

6. Applications

6.1. How fast a colonies population will grow at a certain amount of time.

6.1.1. Using the exponential rule.

6.2. Rate of change of a moving ball.

6.2.1. Power Rule

6.3. Vertical motion of an object

7. Expressions

7.1. f'(x)= lim as h-->0 f(x+h)-f(x)/h

7.2. dy/dt= lim as delta t -->0 f(t + delta t)-f(t)/delta t