Problem 15 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

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

Step-by-Step Solution

Answer

The derivative of the given function \(f(x) = x\) at x = 0 by first principles is 1.

Step 1: Define the function f(x) and first principles formula

The given function is \(f(x) = x\). And we will find the derivative using the first principles (limit) formula: \(f'(x) = \lim_{h \to 0}\dfrac{f(x+h) - f(x)}{h}\)

Step 2: Substitute f(x) and x = 0 in the formula

Now, let's substitute the given function f(x) = x and x = 0 in the first principles formula: \(f'(0) = \lim_{h \to 0}\dfrac{f(0+h) - f(0)}{h} = \lim_{h \to 0}\dfrac{h - 0}{h}\)

Step 3: Simplify the limit

Simplify the limit expression: \(f'(0) = \lim_{h \to 0}\dfrac{h}{h}\) Now, cancel the common factor 'h' in the numerator and the denominator: \(f'(0) = \lim_{h \to 0} 1\)

Step 4: Evaluate the limit

As there is no h left in the expression, the limit becomes a constant value: \(f'(0) = 1\) Hence, the derivative of the given function f(x) = x at x = 0 by first principles is 1.

Key Concepts

Differential Calculus

Differential calculus is a subfield of calculus concerned with the study of rates at which quantities change. It is the mathematical foundation underlying many physical concepts such as velocity, acceleration, and slopes of curves. The core idea in differential calculus is the concept of the derivative, which provides an exact mathematical way to express and calculate how a function changes at any given point.

For instance, if you imagine driving a car, the speedometer shows your rate of change of distance over time - your speed. Differential calculus gives you the tools to model this situation mathematically and find the speed at any instant; that's what a derivative does. In the context of our exercise, the function given is a simple linear function, where the rate of change is constant. This exercise is perfect for beginners as it introduces the fundamental principle of computing derivatives that can be built upon for more complex functions.

Limit Definition of Derivative

The limit definition of the derivative is a mathematical formula used to calculate the derivative of a function at a given point. It is based on the concept of a limit, which in simple terms, refers to what value a function approaches as the input approaches some point.

The expression \(f'(x) = \lim_{h \to 0}\dfrac{f(x+h) - f(x)}{h}\) is the formal definition of the derivative of a function f(x) at the point x. It looks at the function value at a point x and a point slightly away from x, which is x+h. As h gets infinitesimally small, the ratio \(\dfrac{f(x+h) - f(x)}{h}\) represents the slope of the tangent line to the function at x, hence determining the rate of change or the derivative at that point. The step-by-step solution for our exercise involves applying this limit definition to a simple function, taking care to correctly simplify the expression and evaluate the limit.

IIT JEE Mathematics

The Indian Institutes of Technology Joint Entrance Examination (IIT JEE) is a highly competitive test that serves as the gateway for admission to engineering programs at the premier IITs in India. Mathematics is a critical component of the IIT JEE syllabus, testing students on concepts ranging from algebra to calculus. Intensive preparation is required to master these topics.

Understanding the limit definition of a derivative is particularly important for students preparing for the IIT JEE, as it forms the basis for many problems in differential calculus. The exercise we have covered serves as an elementary example of the application of calculus in these exams. As students progress, they will encounter more complex functions and applications of derivatives, making a strong grasp of the basics essential. The problems in the IIT JEE often require a deep understanding of the concepts, quick thinking, and precise calculations, just as we demonstrated in the step-by-step solution for finding the derivative by first principles.