Chain Rule Formula
The chain rule formula is used to find the derivatives of composite functions like d/dx ( (x^{2 }+ 1)^{3}), d/dx (sin 2x), d/dx (ln 5x). The following are a few differentiation formulas. The chain rule formula is used when we have to find the derivative of some functions where "x" in one of the following formulas is being replaced with something else. Let us learn the chain rule formula along with a few solved examples.
 d/dx (x^{n}) = n · x^{n1}
 d/dx (sin x) = cos x
 d/dx (ln x) = 1/x
What Is Chain Rule Formula?
In the previous sections, we saw that the chain rule formula is used to find the derivative of a composite function. There are two forms of chain rule formula.
Formula 1:
Write the given function as a composite function and use the formula:
d/dx ( f(g(x) ) = f' (g(x)) · g' (x)
For example, to find the derivative of d/dx (sin 2x),
express sin 2x = f(g(x)), where f(x) = sin x and g(x) = 2x.
Then by the chain rule formula,
d/dx (sin 2x) = cos 2x · 2 = 2 cos 2x
Formula 2:
We can assume the expression that is replacing "x" to be "u" and apply the chain rule formula.
dy/dx = dy/du · du/dx
For example, to find the derivative of d/dx (sin 2x),
assume that y = sin 2x and 2x = u. Then y = sin u.
By the chain rule formula,
d/dx (sin 2x) = d/du (sin u) · d/dx(2x) = cos u · 2 = 2 cos u = 2 cos 2x
Note: We do not need to remember the chain rule formula. Instead, we can just apply the derivative formulas (which are in terms of x) and then multiply the result by the derivative of the expression that is replacing x.
For example, d/dx ( (x^{2 }+ 1)^{3}) = 3 (x^{2} + 1)^{2} · d/dx (x^{2} + 1) = 3 (x^{2} + 1)^{2} · 2x = 6x (x^{2} + 1)^{2}.
Let us see the applications of the chain rule formula in the following section.
Solved Examples Using Chain Rule Formula

Example 1: Find the derivative of the function y = cos (2x^{2} + 1).
Solution:
To find: The derivative of the given function.
The given function is y = cos (2x^{2} + 1).
Assume that u = 2x^{2} + 1. Then y = cos u.
By the chain rule formula,
dy/dx = dy/du · du/dx
dy/dx = d/du (cos u) · d/dx (2x^{2} + 1)
dy/dx =  sin u · 4x
dy/dx = 4x sin (2x^{2} + 1) (because u = 2x^{2} + 1).
Answer: The derivative of the given function is, dy/dx = 4x sin (2x^{2} + 1).

Example 2: Find the derivative of the function y = \(\sqrt{x^23}\).
Solution:
To find: The derivative of the given function.
The given function is y = \(\sqrt{x^23}\) = (x^{2}  3)^{1/2}
Assume that u = x^{2}  3. Then y = u^{1/2}.
By the chain rule formula,
dy/dx = dy/du · du/dx
dy/dx = d/du (u^{1/2}) · d/dx (x^{2}  3)
dy/dx = 1/2 u^{1/2} · (2x) (by power rule)
dy/dx = x u^{1/2}
dy/dx = x (x^{2}  3)^{1/2}
Answer: The derivative of the given function is, dy/dx = \(\dfrac{x}{\sqrt{x^{2}3}}\).