Question

Explain in detail what a derivative is in mathematics.

Answer 1

A derivative, in mathematics, is a measure of how a function changes with respect to its independent variable. It represents the rate of change or the slope of the function at a given point. The derivative of a function \(f(x)\) with respect to the variable \(x\) is denoted by \(f^{\prime}(\mathrm{x})\) or \(\frac{d f(x)}{d x}\)

To compute the derivative of a function at a specific point, we use the concept of limits. Here's the definition of the derivative:
\[
f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}
\]
In this definition, \(h\) is an infinitesimally small change in the variable \(x\), and the limit represents the value that the expression approaches as \(h\) approaches zero.
Let's consider a simple example. Suppose we have a linear function, \(f(x)=m x+b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. To find the derivative of this function, we apply the definition of the derivative:

\[
\begin{aligned}
f^{\prime}(x) & =\lim _{h \rightarrow 0} \frac{(m(x+h)+b)-(m x+b)}{h} \\
& =\lim _{h \rightarrow 0} \frac{m x+m h+b-m x-b}{h} \\
& =\lim _{h \rightarrow 0} \frac{m h}{h} \\
& =m
\end{aligned}
\]
As expected, the derivative of a linear function is a constant value equal to the slope, \(m\).
Differentiation rules, such as the power rule, product rule, quotient rule, and chain rule, can be used to find the derivatives of more complex functions efficiently.

The derivative has numerous applications, including finding local extrema (maxima and minima) of functions, determining the concavity and inflection points of functions, and solving optimization problems. In addition, derivatives are used in various fields, such as physics (to model velocity and acceleration), engineering, economics, and biology.