Problem 308 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } y=a+b x^{2}, \text { where } a \text { and } b \text { are arbitrary constants, then prove that } x \frac{d^{2} y}{d x^{2}}=\frac{d y}{d x} \text { . } $$

Step-by-Step Solution

Answer

We are given the function \(y = a + bx^2\). To prove that \(x \frac{d^2y}{dx^2} = \frac{dy}{dx}\), we first differentiate the function with respect to \(x\) to find the first derivative: \(\frac{dy}{dx} = 2bx\). Next, we find the second derivative: \(\frac{d^2y}{dx^2} = 2b\). Finally, we verify the given equation: \(x \cdot \frac{d^2y}{dx^2} = 2bx = \frac{dy}{dx}\). Thus, the equation holds true for the given function.

Step 1: First Derivative of y

To find the first derivative of the function \(y = a + bx^2\), we differentiate it with respect to \(x\): \[\frac{dy}{dx} = \frac{d(a + bx^2)}{dx}\] Since \(a\) is a constant, its derivative is 0. Now we'll find the derivative of the \(bx^2\) term: \[\frac{d(bx^2)}{dx} = b \cdot \frac{d(x^2)}{dx} = 2b \cdot x\] Therefore, the first derivative is: \[\frac{dy}{dx} = 0 + 2bx = 2bx\]

Step 2: Second Derivative of y

Now, let's find the second derivative of the function by differentiating the first derivative with respect to x: \[\frac{d^2y}{dx^2} = \frac{d(2bx)}{dx}\] Since \(b\) is a constant, it stays as it is, and now we'll find the derivative of the \(x\) term: \[\frac{d(x)}{dx} = 1\] Therefore, the second derivative is: \[\frac{d^2y}{dx^2} = 2b\]

Step 3: Verification of the given equation

Now that we have both the first and the second derivative, let's verify the given equation \(x \frac{d^2y}{dx^2} = \frac{dy}{dx}\): \[x \cdot \frac{d^2y}{dx^2} = x \cdot (2b) = 2bx\] Given that the first derivative of y is \(\frac{dy}{dx} = 2bx\), the equation holds true: \[x \cdot \frac{d^2y}{dx^2} = \frac{dy}{dx}\] Thus, we have proven that \(x \frac{d^2y}{dx^2} = \frac{dy}{dx}\) for the given function \(y = a + bx^2\), where \(a\) and \(b\) are arbitrary constants.

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus, representing the rate at which a function changes. When dealing with a function of the form \( y = a + bx^2 \), where \( a \) and \( b \) are constants, calculating derivatives helps us understand how the function behaves as \( x \) varies. The derivative of a constant like \( a \) is always zero, which simplifies the process. Meanwhile, for expressions with \( x \), like \( bx^2 \), the power rule of differentiation applies.

• The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \).
• Applying this, the derivative of \( bx^2 \) becomes \( 2bx \).

This gives us the first derivative, \( \frac{dy}{dx} = 2bx \), showing that the rate of change of \( y \) with respect to \( x \) is directly proportional to \( x \). Understanding derivatives is crucial for analyzing the behavior and characteristics of functions.

Proof in Calculus

In calculus, proofs are essential for validating the relationships between different functions and their derivatives. Taking our original function \( y = a + bx^2 \), the task was to prove the equation \( x \frac{d^2y}{dx^2} = \frac{dy}{dx} \). The proof involves several steps, using the principles of differentiation:

• First, find the first derivative, \( \frac{dy}{dx} = 2bx \).
• Next, determine the second derivative by differentiating again: \( \frac{d^2y}{dx^2} = 2b \).

By using these derivatives, we multiply the second derivative by \( x \), which gives \( x \cdot 2b = 2bx \). This matches the first derivative, thus satisfying the given equation.
Through these steps, each application of differentiation directly connects the algebraic expressions of derivatives back to the original function, proving the relationship.

Second Derivative

The second derivative provides insights into the concavity and points of inflection of a function. It is derived from the first derivative by differentiation. For our function \( y = a + bx^2 \), we've already established that the first derivative is \( 2bx \). Differentiating this result gives us a constant second derivative: \( \frac{d^2y}{dx^2} = 2b \).
Several important observations arise from this:

• A constant second derivative implies that the concavity of the function is uniform across its domain.
• When \( 2b > 0 \), the parabola is concave up, and if \( 2b < 0 \), it is concave down.

The second derivative offers powerful information about the nature of the function's graph, helping us understand how it curves and providing a deeper appreciation of the function's landscape. Recognizing the role of the second derivative is key in contexts like optimization and motion analysis in physics.