business calculus drill on derivatives!!!!!!!!!!
TRANSCRIPT
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Business CalculusDrill on Derivatives!!!!!!!!!!
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Derivative Drill & Practice
• I’m going to ask you to remember some derivatives.
• It’s important to be fast as time is money.
• When you think you know the answer,
• (or if you give up ) click to get to the next slide to see if you were correct.
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What’s the definition of f ‘(c)?
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There’s 2 of them!
You really need to know both of these thoroughly!
h
cfhcfh
)()()0lim(
cx
cfxfcx
)()(
)lim(
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What’s the definition of f ‘(x)?
This is a bit different…instead of the limit being a number…it will be a function related to f(x)
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Got it??
Notice that in the second definition, the arrow points to the independent variable in the final answer.
h
xfhxfh
)()()0lim(
( ) ( )lim( )
f x f tx t
x t
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What does the derivative, f ‘(x) tell you about f(x)?
(There are many answers…
how many do you know?)
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Here’s a few:
Slope of the curve
Instantaneous rate of change in f(x)
Rate of change
If f(x) is position…then velocity
If f(x) is velocity…then acceleration
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Now…
Here come a series of questions
concerning basic derivatives you should
know very quickly!
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Ready?
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d/dx(mxn)
• What’s your answer?
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mnxn-1
• Were you right?
(click for next question)
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d/dx(ef(x))
• What is your answer?
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f ’(x) ef(x)
• Were you right?
(click for next question)
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d/dx(ln|x|)
• What is your answer?
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1/x
• Were you right?
(click for next question)
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d/dx(ax)
• What’s your answer…eh?
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(ln a) ax
• Were you right?
(click for next question)
I dig this calculus stuff!!
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d/dx(ln f(x))
•Ready?
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f ‘ (x) / f(x)
• Were you right?
(click for next question)
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d/dx (af(x))
• Go go go go!!!!!
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(ln a) f ‘ (x) af(x)
• Were you right?
(click for next question)
Hit that ball!!
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d/dx(f(x)g(x))
• It’s the Product Rule!
• (don’t flush it!!!)
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f ’(x)g(x) + f(x)g‘(x)
Yeah…I know you know it…but I’m afraid…
Promise me you will remember to use the product rule every time you take the derivative of a
product! OK??
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d/dx(f(x)/g(x))
• It’s the merry mighty Quotient Rule!!
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Pleeeezzze!Don’t get the numerator backwards.
2
'( ) ( ) ( ) '( )
( ( ))
f x g x f x g x
g x
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d/dx(f(g(x)))
Chain chain chain…. ♫
Chain of foooools! ♪ ♫ ♫
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Yeah yeah…you’re good now…
But what about that Chain Rule?
Don’t forget all three steps!1) Derivative of outside 2) Copy inside3) Multiply by the “derivative of inside”
)('))((' xgxgf
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d/dx(y)
Are you confused yet?
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dy/dx
• Did I trick you ?
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Ha ha…I did trick you!
Variable to a variable power - you can’t do it…unless you take the natural log first…
y = xx
ln y = x ln x
(1/y)(dy/dx)=x(1/x)+ 1lnx
dy/dx = y(1+lnx)
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OK…switching to Related Rates
For the rest of the questions, pretend you are doing a related rates problem.
That is…”t” is the independent variable…
For each expression, give the derivative as you would in a Related Rates problem.
Here we go!!!
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y
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dy/dt
Ready to proceed?
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Dr. K is
PROUD of ya!
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Goodbye!
Be sure to run the power points again and again until you are confident with Derivatives.