Derivative of a Function with Respect to Another Function
Derivative of a Function with Respect to Another Function:
 The derivative of a function with respect to another function involves understanding how changes in one function affect the rate of change of another function.
Notation:
 Given two functions $y=f(x)$ and $x=g(t)$, if $y$ depends on $x$ and $x$ depends on $t$, we can denote the derivative of $y$ with respect to $t$ as $\frac{dy}{dt}$.
Chain Rule for Functions of a Single Variable:
 If $y=f(u)$ and $u=g(x)$, then the chain rule states: $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$
Extension to Derivative with Respect to Another Function:
 Given: $y=f(u)$ and $u=g(x)$, also $x=h(t)$
 The derivative of $y$ with respect to $t$ is calculated as follows: $\frac{dy}{dt}=\frac{dy}{du}\cdot \frac{du}{dx}\cdot \frac{dx}{dt}$
Example:
Given $y=\mathrm{sin}(3x)$, $x={t}^{2}+1$, find $\frac{dy}{dt}$.

Calculate $\frac{dy}{dx}$ and $\frac{dx}{dt}$: $\frac{dy}{dx}=3\mathrm{cos}(3x)$ $\frac{dx}{dt}=2t$

Apply the chain rule: $\frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}=3\mathrm{cos}(3x)\cdot 2t=6t\mathrm{cos}(3x)$ Substitute $x={t}^{2}+1$ into the expression: $\frac{dy}{dt}=6t\mathrm{cos}(3({t}^{2}+1))$
Key Considerations:

Sequential Dependency of Functions:
 Ensure the sequential relationship between the functions is welldefined for accurate differentiation.

Chain Rule Application:
 Apply the chain rule successively for functions dependent on multiple variables.
Applications:

Physics and Dynamics:
 Understanding how the rate of change of one variable depends on changes in another variable is crucial in physics, particularly in problems involving multiple forces or interactions.

Economics and Optimization:
 In economics, this concept is applied in optimization problems to analyze the interdependence of variables in models.
Special Cases:

Implicit Functions:
 Derivatives of implicitly defined functions involving multiple variables often require this concept to find the rate of change.

Optimization Problems:
 Problems aiming to optimize one function while considering the dependence on another function utilize this concept for modeling and finding optimal solutions.