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Chain Rule - Derivatives

The chain rule is used to differentiate a function when it is a composition of functions, that is, it contains one function inside another. For example f(x) = sin(x^2 + 2x) is a composition of two functions, the inner function x^2 + 2x and the outer sine function.

The chain rule, in short, says that if for a composition of two functions, in order to differentiate it you need to differentiate both the inner function and the outer function separately and then multiply their results. This will give you the derivative of the original function.

For example, sin(2x^2 + x) contains an inner algebraic expression 2x^2 + x and the outer one is the sine function. Differentiating both separately:
  • d/dx sin(2x^2 + x) = cos(2x^2 + x)
  • d/dx  2x^2 + x = 4x + 1
Now multiply the two derivatives,
cos(2x^2 + x) * (4x + 1) = (4x + 1)cos(2x^2 + x)
So (4x + 1)cos(2x^2 + x) is the derivative of sin(x^2 + 2x).

Examples of some functions on which you would apply the chain rule to find their derivative:

  • f(x) = e^(x^2 + 2x)
  • f(x) = sin(2x)
  • f(x) = e^(-x)
  • f(x) = log(x^2 + 1)
  • f(x) = 1/(x^2 + 1)
  • f(x) = sqrt(x^2 + 1)
In each of the above functions, there are two functions. The following table lists the two functions present in each of the above functions. One is called the inner function and the other is called the outer function.
FunctionInner FunctionOuter Function
e^(x^2 + 2x)x^2 + 2xe^()
sin(2x)2xsin()
e^(-x)-xe^()
log(x^2 + 1)x^2 + 1log()
1/(x^2 + 1)x^2 + 11/()
sqrt(x^2 + 1)x^2 + 1sqrt()

In the above table, the brackets () present in outer functions are the brackets in which the inner functions go in order to form the composite function. 

In order to apply the chain rule properly you need to be able to find the inner and outer functions.

The chain rule formula

Chain Rule - Derivatives
The above definition simply says:
The derivative of a composite function f(g(x)) is given by the product of the derivative of the inner function g(x) with that of the outer function f(u) such that the the value of u = g(x) is substituted after taking the derivative of the outer function.
Again, the above definition is outlined by the following steps:
  • Identify the inner and outer functions
  • Put 'u' in place of the inner function and find its derivative
  • Put back the inner function in place of 'u'
  • Take the derivative of the outer function separately, and multiply it with that of the outer function already taken above
The following example illustrates the application of the chain rule:

Question:
Find the derivative of f(x) = sin(cos(x))
Solution:
  • Identify the inner and outer functions:
    • Inner function: cos(x)
    • Outer function: sin(x)
  • Put 'u' in place of the inner function: sin(u)
  • Evaluate the derivative of the outer function: cos(u)
  • Put back the inner function in place of 'u': cos(cos(x))
  • Take the derivative of the inner function: d/dx cos(x) = -sin(x)
  • Multiply the derivative of the inner function and that of the outer function,
    • Answer: f`(x) = -sin(x) * cos(cos(x))

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