Chain Rule for differentiation

Chain Rule:

Rule:

• For a composite function $f\left(g\left(x\right)\right)$ where $f\left(u\right)$ and $g\left(x\right)$ are differentiable functions, the chain rule states: $\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]={f}^{\mathrm{\prime }}\left(g\left(x\right)\right)\cdot {g}^{\mathrm{\prime }}\left(x\right)$.

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 $f\left(x\right)=\mathrm{sin}\left({x}^{2}\right)$
• Let $u\left(x\right)=\mathrm{sin}\left(x\right)$ and $v\left(x\right)={x}^{2}$
• ${u}^{\mathrm{\prime }}\left(x\right)=\mathrm{cos}\left(x\right)$ and ${v}^{\mathrm{\prime }}\left(x\right)=2x$
• Apply the chain rule: $\frac{d}{dx}\left[\mathrm{sin}\left({x}^{2}\right)\right]=\mathrm{cos}\left({x}^{2}\right)\cdot 2x$

Properties and Notes:

1. Nested Functions:

• The chain rule is applied when functions are nested within each other.
2. Indirect and Direct Compositions:

• It works for both direct compositions ($f\left(g\left(x\right)\right)$) and indirect compositions ($f\left(g\left(h\left(x\right)\right)\right)$).
3. 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 $\mathrm{sin}\left({x}^{2}\right)$ or $\mathrm{cos}\left(3x\right)$.
• Logarithmic and Exponential Functions:

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