Problem 1 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { Given } f(x)=x^{2} \text { , find } f^{\prime}(0), f^{\prime}(1) \& f^{\prime}(-1) \text { by first principles. }\\{\text { Ans. } 0,2,-2\\} $$

Step-by-Step Solution

Answer

Using the first principle of differentiation, the derivatives of the function \(f(x)=x^{2}\) at \(x=0\), \(x=1\), and \(x=-1\) are \(f^{\prime}(0) = 0\), \(f^{\prime}(1) = 2\), and \(f^{\prime}(-1) = -2\), respectively.

Step 1: Understanding the First Principle of Differentiation

The first principle of differentiation states that the derivative of a function \(f(x)\) at a point \(x\) is given by: \[ f^{\prime}(x)=\lim_{{h \to 0}}\frac{f(x+h)-f(x)}{h} \]

Step 2: Substituting the given function into the principle

We substitute \(f(x)=x^{2}\) into the principle of differentiation: \[ f^{\prime}(x) = \lim_{{h \to 0}}\frac{(x+h)^{2} - x^{2}}{h} \] Simplify this equation to get: \[ f^{\prime}(x) = \lim_{{h \to 0}}\frac{x^{2} + 2xh + h^{2} - x^{2}}{h} \] \[ f^{\prime}(x) = \lim_{{h \to 0}}2x + h \]

Step 3: Calculate \(f^{\prime}(0)\)

Substitute \(x = 0\) into the simplified derivative equation: \[f^{\prime}(0) = 2(0) + 0 = 0 \]

Step 4: Calculate \(f^{\prime}(1)\)

Substitute \(x = 1\) into the simplified derivative equation: \[f^{\prime}(1) = 2(1) + 0 = 2 \]

Step 5: Calculate \(f^{\prime}(-1)\)

Substitute \(x = -1\) into the simplified derivative equation: \[f^{\prime}(-1) = 2(-1) + 0 = -2 \]

Key Concepts

First Principle of Differentiation

The first principle of differentiation is a cornerstone concept in calculus. It is a method to find the derivative of a function using the limit process. The derivative represents the rate of change of the function at a particular point. To understand the first principle, let's consider any function \(f(x)\). The idea is to look at the change in \(f(x)\) as we make a small change \(h\) to \(x\). We then evaluate whether this change converges to a particular value as \(h\) approaches zero.

The derivative of the function \(f(x)\) at a point \(x\) is mathematically expressed as:

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

This formula breaks down the function into smaller bits to compute the actual rate at which \(f(x)\) changes with respect to \(x\). This principle is significant because it allows us to compute the slope of the tangent line to the curve at any point \(x\).

To apply this principle, we can substitute our specific function into this formula and perform limit calculations to extract the derivative.

Derivative

A derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It's often described as the function's instantaneous rate of change or slope of the tangent line at any point on its graph. For a given function \(f(x)\), its derivative \(f'(x)\) gives us valuable information.

For example, if \(f(x) = x^2\), the derivative \(f'(x)\) tells us how the output \(f(x)\) will change as \(x\) changes. In the exercise solution, we calculate:

• At \(x = 0\), \(f'(0) = 0\), implying the function does not change at that point.
• At \(x = 1\), \(f'(1) = 2\), indicating an increase at a constant rate of 2.
• At \(x = -1\), \(f'(-1) = -2\), suggesting a decrease at a rate of 2.

These evaluations tell us about the behavior of \( f(x)\) at specific points. By knowing \(f'(x)\), you're looking at everything from the growth to the decline of a function, making derivatives indispensable in real-world applications.

Limit

The concept of a limit is pivotal in calculus, particularly when working with the first principle of differentiation. It analyzes how a function behaves as it approaches a specific point. In the first principle of differentiation, the limit helps us determine the derivative by considering values very close to, but not exactly at, a point.

To calculate the derivative using the first principle, we evaluate the expression \( \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}\). Here, \(h\) symbolizes a minute increment or decrement to the variable \(x\).

By observing how \( \frac{f(x+h) - f(x)}{h} \) behaves as \(h\) approaches zero, we're able to calculate the slope of the tangent line to \(f(x)\). The precise "approaching" behavior is at the core of a limit. It allows us to reach the derivative, a concept giving boundless applications in physics, engineering, and beyond for continuous functions.