BLOG 4. Professor Awad!!

Good afternoon ladies and gentlemen, today Professor Awad will be taking you on an adventure through the study of derivatives.

First off, what in God’s name is a derivative and where in the world do I buy one of these?

Unfortunately for the wealthier students in this class, buying a derivative (unless form Goldman Sachs) is not going to workout today. Lets take a step back before we pinpoint where to get one of these things and understand why you would want them in the first place.

I’m sorry if I painted a much more exciting picture in your minds but all it basically comes down to is the slope of a line at a certain point on the original equation—instead of the derivative equation… don’t worry ill explain. The previously mentioned point on the original equation is also the tangent line; don’t forget that.

Lucky for you guys some really nerdy math geeks figured out a bunch of equations for us to use to find derivatives under different circumstances.

These rules are:

Constant Function- derivative of a constant
example:
y=x
d/dx(x)=0

Linear Function- linear equation (y=mx+b)
example:
y=3/8x+99
d/dx=3/8+0

A function multiplied by a constant: derivative of the formula is multiplied by a constant given to you.

Sum & Difference- functions are being added or subtracted
example:
d/dx[f(x)+g(x)]= d/dx(f(x))+ d/dx(g(x))
d/dx[f(x)-g(x)]=d/dx(f(x))- d/dx(g(x))

The Power Rule- The equation is raised to a power

d/dx(x^n)= nX^n-1

example- d/dx (6x^3)= 18x^2

The Natural Exponent Rule- The equation is a natural number (e) raised to a power

d/dx(e^x)= e^x

example: f(x)= 7e^x-12x---- d/dx=7e^x-(ln12)(12^x)

Derivative of Lnx- Used to find the derivative of a natural log

d/dx (fx)=lnx= 1/x

Product Rule- (f(x) x g(x)) = f(x)` x g(x) + g(x)` x f(x)

Quotient Rule- f(x)/g(x)= f(x)` x g(x) - f(x) x g(x)`/g(x)^2


Never forget that these rules to find the slope pertain to varying structures of the original formula. (i.e. Is there subtraction going on? Multiplication? Addition? Etc.)

Fast Facts

A constant is always Zero.

The power rule= multiplying its original power and derivative power is only one subtracted from the original.

Product rule- the derivative of two functions multiplied is one derivative multiplied by the original of the other added together.

Bang. Bang. Done deal. Now everybody go do your Think Pair Shares, Exit Slips, Group Activities, Weekly quizzes, and surprise pop quiz… before 2:25. MUHAHAHAHA

REAL WOLRD APPLICATION OF ONE OF THE RULES. WHICH RULE AM I USING?

A ball is rolled across a table. It's position is measured by the equation:

s=t^3-3t^2+1, where t represents the time, in seconds, and s represents the current position.

Assuming the ball's speed is always constant, how far does the ball go after 5 seconds? What is the rate of change at 5 seconds?

s(t)=t^3-3t^2+1

s(5)=5^3-3(5)^2+1

s(5)=125-75+1

s(5)=50+1

s(5)=51

s'(t)=3t^2-6t+1

s'(5)=3(5)^2-6(5)+1

s'(5)=75-30+1

s'(5)=45+1

s'(5)=46