Differentiation of the Univariate Functions

Derivatives

A univariate function in mathematics is a function that takes an input variable and produces an output variable. Let’s suppose the position of a car is given by a univariate function $f(x) = (x+2)(x-6)$, where $x$ is time in seconds. In order to calculate the velocity $v(x)$ of the car at any time $x$, we need to compute the instantaneous rate of change of position $f(x)$ with respect to time $x$. For this, we need derivatives.

Given a univariate function $f(x)$, the derivative $\frac{df(x)}{dx}$ represents the instantaneous rate of change (or instantaneous slope) of the function $f$ with respect to $x$. Mathematically, the derivative of a function $f$ at a point $x$ is calculated as follows:

If the derivative $f'(x) < 0$, it means the function is decreasing at that point. Otherwise, it is nondecreasing.

Here are the derivatives of some common functions:

• Constant function: If $f(x) = \text{c}$, where $\text{c}$ is a constant, then $f'(x) = 0$.

• Polynomial function: If $f(x) = x^n$, where $n$ is the degree of the polynomial, then $f'(x) = n \cdot x^{n-1}$ ...