21. Chain rule                                                                                        Chain rule
                                                                                                            Statement
     21.1. Statement                                                                                        Examples
            The power rule says that
                                                d   n      n−1
                                               dx [x ] = nx   .
            This rule is valid for any power n, but not for any base other than the simple input variable
            x. For instance,
                                          d         3            2
                                          dx (2x+5) 6=3(2x+5) .
                                        3                                                                         Table of Contents
            The function h(x) = (2x + 5) is built up of the two simpler functions g(x) = 2x + 5 and
            f(x) = x3:
                                                    3        3
                                     h(x) = (2x+5) = (g(x)) = f(g(x)).                                             ◭◭      ◮◮
            Technically speaking, h is the composition of f and g. The next rule expresses the derivative
            of such a function in terms of the derivatives of its components.                                       ◭      ◮
                     Chain rule. For functions f and g                                                              Page 1 of 8
                                         d [f(g(x))] = f′(g(x)) · g′(x).
                                         dx                                                                           Back
                                                                                                                    Print Version
            In the composition f(g(x)), we call f the outside function and g the inside function. With
            this terminology, the rule says that the derivative of the composition of two functions is              Home Page
            the derivative of the outside function evaluated at the inside function times the derivative
            of the inside function.
     21.2. Examples                                                                                   Chain rule
                                                   d                                                Statement
            21.2.1  Example     Find the derivative   (2x+5)3 .
                                                  dx                                                  Examples
                                               3
            Solution Webegin by viewing (2x+5) as a composition of functions and identifying the
            outside function f and the inside function g. The outside function is the last thing you
            do when computing the expression for a given input x. Here, the outside function is the
            cubing function:
                               3                           3
                        (2x+5) =f(g(x)),    where f(x) = x    and g(x)=2x+5.
            Next, we do the computations required for the chain rule formula:                               Table of Contents
                                             3
                                     f(x) = x              g(x) = 2x+5
                                    f′(x) = 3x2           g′(x) = 2                                          ◭◭     ◮◮
                                  f′(g(x)) = 3(2x+5)2                                                         ◭      ◮
            Finally, we use the formula:
                                d                                                                             Page 2 of 8
                                dx    [f(g(x))]  =    f′(g(x))   ·  g′(x)
                                         ↓               ↓            ↓                                         Back
                                d           3               2
                                dx    (2x+5)     = 3(2x+5)       ·    2                                       Print Version
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                                                   d     5 
            21.2.2  Example     Find the derivative dx sin(x ) .
             Solution  Here, the outside function is the sine function:                                         Chain rule
                            sin(x5) = f(g(x)),  where f(x) = sinx and g(x) = x5.                                Statement
                                                                                                                Examples
             So
                                         f(x) = sinx               g(x) = x5
                                        f′(x) = cosx              g′(x) = 5x4
                                     f′(g(x)) = cos(x5)
             giving
                                      d    [f(g(x))]  = f′(g(x))     ·  g′(x)
                                     dx                                                                               Table of Contents
                                      d      ↓              ↓           ↓
                                           sin(x5)    = cos(x5)      ·   5x4                                            ◭◭     ◮◮
                                     dx
                                                                                                                        ◭       ◮
                                                       d     5 
             21.2.3   Example      Find the derivative     sin x .
                                                       dx                                                                Page 3 of 8
             Solution  Recalling that sin5 x means (sinx)5, we see that the outside function is the one
             that raises an input to the fifth power:                                                                       Back
                               5                                5                                                       Print Version
                            sin x = f(g(x)),   where f(x) = x      and g(x)=sinx.
             In order to reduce the number of steps, we go immediately to the chain rule formula and                     Home Page
             do the intermediate computations mentally as required:
                                           d     [f(g(x))]    = f′(g(x))       ·   g′(x)                                         Chain rule
                                          dx
                                           d        ↓                ↓             ↓                                           Statement
                                                   sin5 x     = 5(sinx)4       ·   cosx                                          Examples
                                          dx
                                                                d h 2         i
               21.2.4    Example        Find the derivative dx 5x −4x+3 .
               Solution   Here, the outside function is the exponential function with base 5:
                            x2−4x+3                                    x                    2                                           Table of Contents
                           5          =f(g(x)),     where f(x) = 5         and g(x)=x −4x+3.
               Trimming the number of steps a bit more, we omit the formula for the chain rule and                                        ◭◭       ◮◮
               just think “Derivative of outside function, evaluated at inside function, times derivative of
               inside function”:              h           i
                                            d     2              2                                                                         ◭       ◮
                                                 x −4x+3        x −4x+3
                                           dx 5             =5           ln5·(2x−4).
                                                                                                                                           Page 4 of 8
                                                                d √ x          3
               21.2.5    Example        Find the derivative dx        5e +4x .
                                                                                                                                              Back
               Solution   The outside function is the square root function:
                                                                                                                                          Print Version
                                         h             i
                                      d   p x         3     1    x      3 −1/2     x       2
                                     dx     5e +4x = 2(5e +4x )                (5e +12x ).
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