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Chain Rule

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Derivation

When a function has another function as its argument, another rule is necessary to calculate this derivative. That rule is called the Chain Rule. Common functions that take another function as an argument are: the goniometric functions, the logarithm, and exponential functions. The Chain Rule can be written as shown below:

Let the function h(x)h(x) consist of the function f(x)f(x) with x equal to the function g(x)g(x):

h(x)=f(g(x))\begin{equation} h(x) = f(g(x)) \end{equation}

Then the derivative h(x)h'(x) is equal to:

h(x)=f(g(x))g(x)\begin{equation} h'(x) = f'(g(x))*g'(x) \end{equation}

The derivative h(x)h'(x) of a function h(x)h(x) that consists of a function of which the argument is another function g(x)g(x), is equal to the derivative of that function with the same argument f(g(x))f'(g(x)) times the derivative of the argument g(x)g'(x).

Example

The derivative h(x)h'(x) of the function h(x)h(x) shown below must be calculated. It can be easily seen that the function consists of a function with another function as its argument.

h(x)=sin(4x2)\begin{equation} h(x) = sin(4x^2) \end{equation}

In this case the function h(x)h(x) consists of the functions f(x)=sin(x)f(x) = sin(x) and g(x)=4x2g(x) = 4x^2 thus the Chain Rule must be applied. The working out of the Chain Rule is shown below.

  1. f(x)=sin(x)f(x) = sin(x) and g(x)=4x2g(x) = 4x^2 (The different parts of the Chain Rule are identified)
  2. h(x)=cos(4x2)8xh'(x) = cos(4x^2)*8x (The Chain rule is used with the standard derivation rules)
  3. h(x)=8xcos(4x2)h'(x) = 8xcos(4x^2) (The answer is written neatly)

Now is shown what the Chain Rule is and when it must be used. An example of the Chain Rule is also worked out. The most common rules of derivation are now explained and with these set of rules most of the possible derivatives can be easily found.