Rules of Differentiation

1. Example:

2. Product Rule: Used to find the derivative of the product of two or more functions. The formula is (f*g)'=f'*g+g'f

2.1. Example: f(x)=x^3lnx. We apply the formula which would give us the following function x^3*1/x+3x^2+lnx. Then we simplify and factor if necessary. The answer is x^2(1+3lnx)

3. Derivatives of Natural Logarithms: A tool for finding derivatives. You can convert most functions into logarithmic functions by adding Ln and following the process shown in the example.

4. Higher Order Derivatives: There are some functions that when differentiated one time, they're still functions. Therefore, they may be differentiated until there is no longer a function i.e. when we reach zero. An important example of useful HOD are Position, Velocity, Acceleration Functions.

5. Implicit Differentiation: the process of finding the derivative of a dependent variable in an implicit function by differentiating each term separately, by expressing the derivative of the dependent variable as a symbol, and by solving the resulting expression for the symbol. From http://www.merriam-webster.com/dictionary/implicit%20differentiation

5.1. Example:

6. Chain Rule: Used to differentiate different compositions of functions. Process: f(x)= (n(u)^n-1)*the derivative of u

7. Related Rates: Method used to identify unknown rates of change of movement with other rates of change. Process:1)Identify what rate of change you're being asked for. 2) Once you have identified the rate equal it with D(variable of the rate)/DX. Also identify any other information given f.e. radius, diameter, height. 3) Identify the equation needed to find the rate. F.E. the area of a circle= pi*r^2 or the volume of a sphere 4/3PiR^4) Find the derivative of the equation and substitute the values. 5) Simplify.

7.1. Example:  When a circular shield of bronze is heated over a fire its radius increases at the rate of 1/5 cm/sec. At what rate is the shield's area increasing when the radius is 50 cm? First we identify what rate we want to find which is the rate of change of the area. Then we equal it with dA/dt. We also identify the rate of 15cm/sec and at the size they're asking us to find the rate of change. After that, we see that we need the area formula of a circle. We begin to start the equation: PiR^2DR/DT. We make a derivative so we have Pi 2R DR/DT. We substitute the values with Pi 2(50) (1/5). After simplifying we find that dA/dt equals 20pi cm/s. From: http://www.mathscoop.com/calculus/derivatives/applications/related-rates.php

8. Quotient Rule: Used to find the derivative of the quotient of two or more functions. The formula is (f/g)'= (g'*f-f'*g)/(g)^2

8.1. Example: f(x)=2/x+1. F'(x)= (1)(2)-(0)(x+1)/(x+1)^2. After simplifying -2/(x+1)^2=f'(x)

9. Trigonometric Functions are solved according to each individual functions. Multiple rules including product, chain and quotient rule may be used along with trig identities, Below are some examples

9.1. Derivatives of trigonometric functions:

10. Exponential Differentiation are solved the following way: f(x)=e^nx then f’(x)=n’e^nx

10.1. Example: y=2e^2x+1 y’= 2e^2x+1