Transcript Review of Derivatives - PowerPoint Presentation

Review of Derivatives
Power rule, product rule, quotient
rule and chain rule
The Power Rule
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Remember, the power rule only works
on functions of the form y = xn.
The power rule says that y’ = nxn-1
Examples:
y = x2, so y’ = 2x
 y =x1/2, so y’ = ½x-1/2
 y = x -1, so y’ = -x -2
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The Product Rule
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The product rule can be used when two
functions are multiplied together.
If y = f(x)g(x), then y’ = f’(x)g(x) +
f(x)g’(x)
Examples:
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If y = xsinx, then y’ = sinx +xcosx
If y = (3x)(5x+1), then y’ = (3)(5x+1) + (3x)(5)
Of course, you must remember to simplify
your answers!
The Quotient Rule
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The quotient rule can be used when two
functions are being divided.
If y = f(x)/g(x), then
y’ = [g(x)f’(x) – f(x)g’(x)]/(g(x))2,
or
(lodhi – hidlo) / lolo ! 
Example:
If y = sinx/cosx, then y’ = [cosx(cosx) – sinx(sinx)]/cos2x
 What does this simplify to???
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Trigonometric Derivatives
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If
If
If
If
If
If
y
y
y
y
y
y
=
=
=
=
=
=
sinx, then y’ = cosx
cosx, then y’ = -sinx
tanx, then y’ = sec2x
secx, then y’ = secxtanx
cscx, then y’ = -cscxcotx
cotx, then y’= -csc2x
The Chain Rule
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The chain rule is used on composition
functions.
You must identify the inside function
and the outside function.
The chain rule says if y = f(g(x), then
y’ = f’(g(x))*g’(x), or the derivative of
the inside times the derivative of the
outside
The Chain Rule (cont’d)
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Examples:
If y = sin(x2), then y’ = 2xcos(x2)
 If y = (2x+1)3 then y’ = 2*3(2x+1)2
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Remember, the product rule and the
quotient rule may also need to be
used along with the chain rule!!
If y = (2x+1)3(3x+2)2, then y’ =
2*3(2x+1)2(3x+2)2 + (2x+1)3(3)(2(3x+2))
 Don’t forget to simplify!!!
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