Chain Rule Formula

Chain Rule Formula

In differential calculus, we come across the chain rule. It is implemented to determine a composite function’s derivative value. It is an important formula in differentiation.

Say y = f(h(x)), then using the chain rule we get the instantaneous rate of transformation of this function ‘f’ in respect to ‘h’ whereas ‘h’ relative to x gives a differential coefficient of change of the function ‘f’ relative to x. Therefore, we have the derivative: y’ = f’(h(x)). h’ (x). 

Let f depict a function which further comprises of two different functions m and n such that:

f = m(n(x))

Let us assume n(x) = r

So if the functions m and n are differentiable, also if dr/dx and dn/dt exist, then the given composite function in this problem; f(x) is bound to be differentiable. The process has been elaborated in detailed steps below:

We can jot down the differentiation for the given composite function by applying Leibnitz notation. This is done as:

df/dx = (dn/dr) x (dr/dx)

As one can guess from the name, the chain rule infers that each term needs to be differentiated one after another in a chain format. The process begins with the outermost function and ends with the innermost function. 

In simple words, to differentiate or to determine the instantaneous rate of change of a composite function f at any point within its domain, you have to first differentiate its outer portion followed by multiplication with the next inner function’s derivative. This gives us the desired derivative.

What is chain rule formula?

The chain rule formula set for a function y = f(x), is given as dy/dx = dy/dm. dm/dx. 

Here, f(x) is the composite function where x = g(t).

Standard formula for chain rule: dy/dx = dy/dm. dm/dx

Solved Examples

1. g(x) = -2x + 5

f(x) = 6x + 3. Using the chain rule, calculate a’(x) = f (g(x))

Solution: The derivatives of g and f are calculated:

g’(x) = -2

f’(x) = 6

As per the chain rule, 

a’(x) = f’(g(x)) g’(x)

= f’ (-2x + 5). (-2)

= (-2) 6 = -12

As the functions are linear, the solution is trivial. 

Answer: -12

2. f(x) = e^x

g(x) = 6x

Using the chain rule, calculate a’(x), where a(x) = f(g(x))

Solution: Derivative of an exponential function that has base e, is the function itself. 

Therefore, f’(x) = e^x

g’(x) = 6

Applying the chain rule, a’(x) = f’(g(x)) g’(x)

Or, a’(x) = f’ (6x). 6 = 6e^6x

Here, it is crucial to find the derivate at 6x.

Derivative of the function a(x) = f(g(x)) = e^6x does not match with 6e^6x

The only right solution is a’(x) = 6e^6x