## 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}