# 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 $y=f\left(x\right)$ and $x=g\left(t\right)$, 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 $y=f\left(u\right)$ and $u=g\left(x\right)$, then the chain rule states: $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$

### Extension to Derivative with Respect to Another Function:

• Given: $y=f\left(u\right)$ and $u=g\left(x\right)$, also $x=h\left(t\right)$
• 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 $y=\mathrm{sin}\left(3x\right)$, $x={t}^{2}+1$, find $\frac{dy}{dt}$.

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

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

### Key Considerations:

• Sequential Dependency of Functions:

• Ensure the sequential relationship between the functions is well-defined 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.