Derivative of a function with respect to another function

Derivatives with respect to another function:

When taking the derivative of a function with respect to another function, we use the chain rule, treating the outer function as the variable and the inner function as the function.
This process is also known as implicit differentiation, and is often used to find the slope of a curve at a specific point or to solve equations where the dependent variable is not explicitly given in terms of the independent variable.
For example, if we have y = f(g(x)), then we can find dy/dx by applying the chain rule: dy/dx = (dy/dg) * (dg/dx).

The second order derivative of a function is the derivative of its first order derivative. It measures how the rate of change of a function changes as the input value changes.
The second order derivative is denoted by f''(x) or d^2y/dx^2, and is calculated by taking the derivative of the first order derivative of the function.
If the second order derivative is positive at a certain point, then the function is said to be concave up at that point, whereas if it is negative, the function is said to be concave down.
A point where the second order derivative is zero is known as an inflection point, where the function transitions from being concave up to concave down, or vice versa.
Second order derivatives can be used to find the maximum or minimum points of a function, as a maximum or minimum occurs at a point where the second order derivative changes sign.
In some cases, the second order derivative can be used to determine the behavior of the function at extreme values, such as asymptotes or singularities.
The second order derivative can be extended to functions of multiple variables, where it measures the rate of change of the gradient of the function.

$$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx} = f'(u)\cdot g'(x)$$

Second Derivative of a Function: $$\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)$$

Product Rule for Second Derivatives: $$\frac{d^2}{dx^2}(f(x)g(x)) = f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x)$$

Chain Rule for Second Derivatives: $$\frac{d^2y}{dx^2} = \frac{d^2y}{du^2}\cdot\left(\frac{du}{dx}\right)^2 + \frac{dy}{du}\cdot\frac{d^2u}{dx^2}$$