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picture1_Chain Rule Pdf 171538 | Der Item Download 2023-01-26 20-57-02


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File: Chain Rule Pdf 171538 | Der Item Download 2023-01-26 20-57-02
chain rule short cuts in class we applied the chain rule step by step to several functions here is a short list of examples 1 powers of functions the rule ...

icon picture PDF Filetype PDF | Posted on 26 Jan 2023 | 2 years ago
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                                                      Chain Rule Short Cuts
                   In class we applied the chain rule, step-by-step, to several functions. Here is a short list
                   of examples.
                   1. Powers of functions
                   The rule here is
                                                       d       a          a−1 ′
                                                       dxu(x) =au(x)         u(x)                                  (1)
                   So if
                                                                              5
                                                          f(x) = (x+sinx) ,
                   then
                                                   f′(x) = 5(x+sinx)4(1+cosx).
                   The rule (1) is useful when differentiating reciprocals of functions. If a = −1 we get
                                                           d   1   =−u′(x).
                                                                             2
                                                          dxu(x)        u(x)
                   You could also have derived this using the quotient rule.
                   2. Exponentials
                   For a > 0,
                                                       d u(x)       u(x)  ′
                                                      dxa       =a      u(x) lna,                                  (2)
                   So if
                                                                       2
                                                             g(x) = 3x −4x,
                   then
                                                                2
                                                     g′(x) = 3x −4x(2x−4) ln3.
                   3. The Natural logarithm of a function
                   The chain rule in this case says that
                                                         d lnu(x) =      1 u′(x)                                   (3)
                                                        dx             u(x)
                   So if
                                                           f(x) = ln(sinx),
                   then                                              1
                                                          f′(x) = sinx cosx.
              4. Trigonometric functions
              We’ll illustrate the chain rule with the cosine function.
                                       d cosu(x) = −sinu(x)u′(x)                    (4)
                                      dx
              Thus, if
                                                        3
                                          ψ(x) = cos(1+x ),
              then
                                         ′       2        3
                                       ψ(x)=−3x sin(1+x ).
              Functions of the form sinu(x) and tanu(x) are handled similarly.
              5. Inverse trigonometric functions
              We’ll use the arctan function. The chain rule tells us that
                                     d arctanu(x) =   1    u′(x).                   (5)
                                                         2
                                     dx            1+u(x)
              So if
                                        ϕ(x) = arctan(x+lnx),
              then                               1         1
                                    ϕ′(x) =           2  1+    .
                                           1+(x+lnx)        x
              Functions of the form arcsinu(x) and arccosu(x) are handled similarly.
                Bear in mind that you might need to apply the chain rule as well as the product and
              quotient rules to to take a derivative. You might also need to apply the chain rule more
              than once. For example,
                           d             2               2    1
                             sin(ln(x−2x )) = cos(ln(x−2x ))     2 (1 − 4x).
                           dx                              x−2x
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...Chain rule short cuts in class we applied the step by to several functions here is a list of examples powers d dxu x au u so if f sinx then cosx useful when dierentiating reciprocals get you could also have derived this using quotient exponentials for dxa lna g ln natural logarithm function case says that lnu dx trigonometric ll illustrate with cosine cosu sinu thus cos sin form and tanu are handled similarly inverse use arctan tells us arctanu lnx arcsinu arccosu bear mind might need apply as well product rules take derivative more than once example...

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