Definition of the Derivative

Hello JEE aspirants! Welcome to the fascinating world of derivatives. In JEE Main, derivatives form the bedrock of calculus problems, appearing directly and indirectly in various questions. Understanding the definition of the derivative is crucial for mastering applications like tangents, normals, maxima, minima, and more. Let's dive in!

Conceptual Explanation

Imagine you're driving on a curved road. At any instant, the speedometer shows your instantaneous speed. The derivative is essentially a mathematical speedometer – it tells you the instantaneous rate of change of a function at a specific point. Geometrically, it represents the slope of the line tangent to the curve at that point.

Think about a secant line cutting through a curve at two points. As these two points get closer and closer, the secant line approaches the tangent line. The derivative is the limit of the slope of this secant line as the distance between the points approaches zero. For example, consider the function $f(x) = x^2$ . The slope between two points, say $x$ and $x+h$ , is given by $\frac{(x+h)^2 - x^2}{h}$ . As $h$ becomes infinitesimally small, this ratio approaches the derivative of $x^2$ .

Derivative as a Limit

Formally, the derivative of a function $f(x)$ at a point $x$ is defined as the limit:

f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}

This formula calculates the instantaneous rate of change of $f(x)$ with respect to $x$ . Consider $f(x) = \sin(x)$ . Then, $f(x+h) = \sin(x+h)$ . Using the formula, we try to evaluate $lim_{h \rightarrow 0} \frac{sin(x+h) - sin(x)}{h}$ , which will eventually lead us to $cos(x)$ .

An alternative, but equivalent, definition of the derivative at a point $a$ is:

f'(a) = \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}

This formula helps find the derivative at a specific point $x = a$ . Suppose you want to find the derivative of $f(x) = x^3$ at $x = 2$ . Using this formula, we have $lim_{x \rightarrow 2} \frac{x^3 - 2^3}{x - 2}$ , which simplifies to $12$ .

Geometric Interpretation

The derivative $f'(x)$ gives the slope of the tangent to the curve $y = f(x)$ at the point $(x, f(x))$ . This is a crucial concept for solving problems involving tangents and normals.

If $f'(x) > 0$ , the function is increasing at $x$ . If $f'(x) < 0$ , the function is decreasing at $x$ . If $f'(x) = 0$ , the function has a stationary point at $x$ (which could be a local maximum, local minimum, or point of inflection).

Left-Hand Derivative (LHD)

The Left-Hand Derivative (LHD) at a point $a$ is defined as:

LHD \text{ at } a = \lim_{h \rightarrow 0^-} \frac{f(a+h) - f(a)}{h}

Here, $h$ approaches 0 from the negative side (values less than 0). The LHD essentially looks at the slope of the function just to the left of the point $a$ . For example, to find the LHD of $|x|$ at $x=0$ , we analyze the behavior of $|x|$ just to the left of 0. Since $|x| = -x$ for $x<0$ , the LHD becomes -1.

Right-Hand Derivative (RHD)

The Right-Hand Derivative (RHD) at a point $a$ is defined as:

RHD \text{ at } a = \lim_{h \rightarrow 0^+} \frac{f(a+h) - f(a)}{h}

Here, $h$ approaches 0 from the positive side (values greater than 0). The RHD looks at the slope of the function just to the right of the point $a$ . For instance, for the same function $|x|$ at $x=0$ , we analyze the behavior to the right of 0. Since $|x| = x$ for $x>0$ , the RHD becomes 1.

Condition for Differentiability

A function $f(x)$ is differentiable at a point $x = a$ if and only if:

The LHD at $a$ exists and is finite.
The RHD at $a$ exists and is finite.
LHD at $a$ = RHD at $a$ .

In simpler terms, the function must have a unique tangent at that point. Consider the modulus function $f(x) = |x|$ at $x = 0$ . We found earlier that the LHD at 0 is -1, and the RHD at 0 is 1. Since LHD ≠ RHD, the modulus function is not differentiable at $x = 0$ , even though it is continuous.

Tip: Always check for LHD and RHD when dealing with piecewise functions or functions involving modulus, greatest integer, or fractional part. These functions often have points where they are not differentiable.

Common Mistake: Assuming continuity implies differentiability. A function can be continuous at a point but not differentiable at that point (e.g., $f(x) = |x|$ at $x = 0$ ).

JEE-Specific Tricks

For JEE Main, remember these tricks:

If a function is differentiable at a point, it is necessarily continuous at that point. However, the converse is not always true.
When dealing with limits involving derivatives, L'Hôpital's Rule can often simplify the calculations.

For example, consider a question where you're asked to find the value of a constant that makes a piecewise function differentiable at a certain point. You'll need to equate the LHD and RHD at that point and solve for the constant.

Mastering the definition and conditions of differentiability is crucial for cracking JEE Main. Keep practicing, and you'll ace it!