Introduction
Differentiation is the process of finding the derivative of a function, which we studied in the chapter on differentiability. There, we defined the derivative as the limit of the difference quotient and understood it as the rate of change of the function. In this chapter, we will focus on techniques to compute derivatives quickly and efficiently, without relying on the first-principles definition every time.
We will learn standard differentiation rules that make the process straightforward, such as the power rule, product rule, quotient rule, and chain rule. These rules allow us to differentiate functions systematically, avoiding the need to evaluate limits repeatedly. We will also explore higher-order derivatives and their significance in analyzing the behavior of functions.
By the end of this chapter, we will be able to differentiate a wide range of functions, including polynomials, trigonometric functions, exponential functions, and logarithmic functions, using a set of well-defined rules and techniques.
Derivative
Let us recall the general definition of the derivative:
where \( x \) is a general point in the domain of the function. The function is said to be differentiable at \( x \) if this limit exists. If the limit does not exist at some point, then \( f'(x) \) does not exist at that point.
We know that \( f'(x) \) represents the slope of the tangent to the curve \( y = f(x) \) at the point \( (x, f(x)) \). If the function is differentiable at \( x \), then the graph of \( y = f(x) \) has a well-defined tangent line at \( (x, f(x)) \), and its slope is given by \( f'(x) \).
Differentiability implies continuity, meaning that if \( f(x) \) is differentiable at \( x \), then \( f(x) \) must also be continuous at \( x \). However, the converse is not necessarily true; a function can be continuous at a point but not differentiable there. This happens, for example, at points where the graph has sharp corners or cusps.
In this chapter, we will learn methods to compute derivatives efficiently, using differentiation rules that allow us to find \( f'(x) \) without directly evaluating the limit definition each time. But before we do that let us have a look at another interpretation of derivative.
Derivative as Rate of Change
The derivative of a function can be interpreted as the rate of change of the output with respect to the input. Consider a function \(y = f(x)\), where \(x\) is the independent variable and \(y\) is the dependent variable. If the input \(x\) is increased by a small amount \(\Delta x\), then the output also changes, leading to a corresponding change \(\Delta y\). The new output is given by
The change in the function value, \(\Delta y\), is given by
Comparing this with the definition of the derivative in previous sections, \(h\) in the difference quotient corresponds to \(\Delta x\). Thus, the formulation in terms of \(\Delta x\) is consistent with the previous definition of the derivative.
The average rate of change of \(y\) with respect to \(x\) over the interval \([x, x + \Delta x]\) is defined to be
The average rate of change measures how much the function value changes per unit change in \(x\) over a finite interval. This expression represents the slope of the secant line joining the points \((x, f(x))\) and \((x + \Delta x, f(x + \Delta x))\) on the graph of \(y = f(x)\). The secant line provides a linear approximation of the function over that interval, but it does not capture instantaneous behavior.
The instantaneous rate of change is obtained by letting \(\Delta x \to 0\), giving the limiting value:
This limit, if it exists, defines the derivative of \(f(x)\) at \(x\) and represents the slope of the tangent line to the curve at \((x, f(x))\). Unlike the secant line, which depends on a finite interval, the tangent line captures the precise behavior of the function at a single point.
The derivative is denoted by
The derivative provides a rigorous mathematical formulation of instantaneous rate of change. The notation \(\frac{dy}{dx}\) is called Leibniz notation, named after Gottfried Wilhelm Leibniz.
Some Physics
One can better understand the concept of the derivative as a rate of change using a physical example of an object moving in a straight line. Let \( x \) represent the position of the object at time \( t \), where \( x \) is a function of \( t \), written as \( x = f(t) \). As time \( t \) changes, the position \( x \) also changes.
The change in time over a certain interval is denoted by \( \Delta t \), and the corresponding change in position during that time is given by \( \Delta x = f(t + \Delta t) - f(t) \). The average velocity over the time interval \( [t, t + \Delta t] \) is defined as
This represents the average rate of change of position over the interval \( [t, t + \Delta t] \) and corresponds to the slope of the secant line joining the points \( (t, f(t)) \) and \( (t + \Delta t, f(t + \Delta t)) \) on the graph of \( x = f(t) \).
To obtain the instantaneous velocity, the time interval is made arbitrarily small by taking the limit as \( \Delta t \to 0 \):
This limit, when it exists, gives the instantaneous rate of change of position with respect to time and is called the instantaneous velocity of the object. It is denoted by
The instantaneous velocity represents the exact speed and direction of the object at a specific moment in time. Geometrically, it corresponds to the slope of the tangent line to the position-time graph at \( (t, f(t)) \).
Thus, the derivative provides a precise way to measure how rapidly a quantity changes, whether it is position with respect to time (velocity), temperature with respect to depth, or any other physical quantity that varies continuously.