Chain Rule for differentiation
Chain Rule:
Rule:
 For a composite function $f(g(x))$ where $f(u)$ and $g(x)$ are differentiable functions, the chain rule states: $\frac{d}{dx}[f(g(x))]={f}^{\mathrm{\prime}}(g(x))\cdot {g}^{\mathrm{\prime}}(x)$.
Explanation:
 The chain rule allows us to find the derivative of a function within a function. It considers the rate of change at one function nested within another and combines these rates to find the overall rate of change.
Example:
 Consider $f(x)=\mathrm{sin}({x}^{2})$
 Let $u(x)=\mathrm{sin}(x)$ and $v(x)={x}^{2}$
 ${u}^{\mathrm{\prime}}(x)=\mathrm{cos}(x)$ and ${v}^{\mathrm{\prime}}(x)=2x$
 Apply the chain rule: $\frac{d}{dx}[\mathrm{sin}({x}^{2})]=\mathrm{cos}({x}^{2})\cdot 2x$
Properties and Notes:

Nested Functions:
 The chain rule is applied when functions are nested within each other.

Indirect and Direct Compositions:
 It works for both direct compositions ($f(g(x))$) and indirect compositions ($f(g(h(x)))$).

Multiple Functions in Sequence:
 The chain rule extends to more complex functions where multiple functions are nested within each other.
Applications of the Chain Rule:

Trigonometric Functions:
 Used in differentiation of compositions involving trigonometric functions, such as $\mathrm{sin}({x}^{2})$ or $\mathrm{cos}(3x)$.

Logarithmic and Exponential Functions:
 Applied in finding derivatives of compositions involving logarithmic and exponential functions, like ${e}^{2{x}^{2}}$ or $\mathrm{ln}({x}^{3})$.