Derivatives

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Derivatives

• By Mendy Kahan Jared Friedman

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What is a Derivative?

• Lets say we were given some function called f
  and the derivative of that function is f. f
  represents the slope of the tangent line to the
  graph at point x.
• This is represented by

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How Do We Take Derivatives?
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Chain Rule
This rule should be used for difficult problems
involving composite functions
The derivative of this composition is given by
the formula

• This can also be written as

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Chain Rule Examples

• For more Examples
• http//web.mit.edu/wwmath/calculus/differentiation
  /chain.html
• http//www.ugrad.math.ubc.ca/coursedoc/math100/not
  es/derivative/chain.html
• http//www.exambot.com/cgi/topic/show.cgi/math/dif
  c/diff/dcha

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Product Rule
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Product Rule Examples
For more Examples http//www.math.ucdavis.edu/ko
uba/CalcOneDIRECTORY/productruledirectory/ProductR
ule.html
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Quotient Rule



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Quotient Rule Examples
For more Examples http//www.math.hmc.edu/calculu
s/tutorials/quotient_rule/
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High Order Derivatives
The rules for taking higher order derivatives are
basically identical to taking a regular
derivative. Once you take the first derivative of
the answer repeat this process as many times as
necessary to find the nth derivative. However,
there are sometimes patterns and it would be very
time consuming to take higher order derivatives.
Ex

For more examples http//www.exambot.com/cgi/topi
c/show.cgi/math/difc/diff/hder
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Higher Order Derivative Ex

• Find the 7th derivative

1. The pattern repeats every 4th derivative so

1. Find the pattern of the derivatives of sin x

Since the remainder is 3, the 7th derivative of
sin x is the same as the 3rd derivative.
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Derivatives of Trigonometric Functions
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Trigonometric Function Example
For more examples http//www.sosmath.com/calculus
/diff/der03/der03.html
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Derivatives of Exponential Logarithmic Functions

• Rules

Ex
For more examples http//www.exambot.com/cgi/prob
lem/show.cgi/math/difc/diff/dcha/1166.pr?a
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The End

• Now you are ready to take the test!!!