Problem 269 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } y=x^{3} \ln x, \text { find } \frac{d^{4} y}{d x^{4}} $$

Step-by-Step Solution

Answer

The fourth derivative of the given function is: \(\frac{d^4 y}{d x^4} = \frac{6}{x}\).

Step 1: Chain Rule

If a function is composed of two or more functions, such as \(u(x)v(x)\), the chain rule states that the derivative of the composite function is the product of the derivatives of the inner and outer functions: \(\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)\).

Step 2: Power Rule

If a function has the form \(f(x) = x^n\), where n is a constant, the power rule says that the derivative of f(x) with respect to x is given by: \(f'(x) = nx^{n-1}\). **Step 2: First derivative**

Step 3: Apply product rule

We have: \(y = x^3 \ln(x)\). To find the first derivative, apply the product rule: \( \frac{dy}{dx} = (x^3)' \ln(x) + x^3 (\ln(x))' \)

Step 4: Find individual derivatives

Using the power rule: \((x^3)' = 3x^2\). Now, for the derivative of \(\ln(x)\), we have \((\ln(x))' = \frac{1}{x}\).

Step 5: Combine results

Combining our results from the previous steps, we get: \( \frac{dy}{dx} = 3x^2 \ln(x) + \frac{x^3}{x} = 3x^2 \ln(x) + x^2 \) **Step 3: Second derivative**

Step 6: Apply product rule

Now, we will apply the product rule to the first derivative \( \frac{dy}{dx} = 3x^2 \ln(x) + x^2\): \( \frac{d^2y}{dx^2} = (3x^2 \ln(x))' + (x^2)' \)

Step 7: Find individual derivatives

Using the power and product rules: \((3x^2 \ln(x))' = (3x^2)' \ln(x) + 3x^2 (\ln(x))' = 6x \ln(x) + \frac{3x^2}{x} = 6x \ln(x) + 3x\) \((x^2)' = 2x\)

Step 8: Combine results

Combining our results from the previous steps, we get: \( \frac{d^2y}{dx^2} = 6x \ln(x) + 3x + 2x = 6x \ln(x) + 5x \) **Step 4: Third derivative**

Step 9: Apply product rule

Now, we will apply the product rule to the second derivative \( \frac{d^2y}{dx^2} = 6x \ln(x) + 5x \): \( \frac{d^3y}{dx^3} = (6x \ln(x) + 5x)' \)

Step 10: Find individual derivatives

Using the power and product rules: \((6x \ln(x))' = (6x)' \ln(x) + 6x (\ln(x))' = 6(\ln(x) + 1)\) \((5x)' = 5\)

Step 11: Combine results

Combining our results from the previous steps, we get: \( \frac{d^3y}{dx^3} = 6(\ln(x) + 1) + 5 \) **Step 5: Fourth derivative**

Step 12: Apply product rule

Now, we will apply the product rule to the third derivative \( \frac{d^3y}{dx^3} = 6(\ln(x) + 1) + 5 \): \( \frac{d^4y}{dx^4} = (6(\ln(x) + 1) + 5)' \)

Step 13: Find individual derivatives

Using the power and product rules: \((6(\ln(x) + 1) + 5)' = 6(\ln(x) + 1)' = 6 \cdot \frac{1}{x}\)

Step 14: Final result

The fourth derivative of the given function is: \(\frac{d^4 y}{d x^4} = \frac{6}{x}\)

Key Concepts

Chain Rule

The chain rule is essential for finding the derivative of composite functions. Imagine you have two functions nested within each other, like a Russian doll. The chain rule helps us solve for the derivative of such functions efficiently. For instance, if you have a function expressing another function inside it, such as \(u(x)v(x)\), the chain rule tells us to take the derivative of the outer function with respect to the inner one while multiplying by the derivative of the inner function.

• This concept is especially useful when dealing with logarithmic, exponential, and trigonometric functions within polynomials.
• Misunderstanding the chain rule can lead to incorrect results, especially in complex differentiation problems.

To apply it effectively, practice by breaking down composite functions into simpler parts that you can differentiate one step at a time.
Remember, keeping track of each step is crucial to mastering the application of the chain rule.

Power Rule

The power rule simplifies the process of differentiation for polynomial terms. Whenever you see a function in the form of \(x^n\), the power rule allows you to quickly find its derivative. Simply multiply the exponent, \(n\), by the coefficient of \(x\), and then reduce the exponent by one. For example, the derivative of \(x^3\) is \(3x^2\).

• Easy to memorize, this rule forms the backbone of most differentiation tasks.
• It's usually your go-to method whenever you encounter polynomial expressions in differentiation problems.

One important thing to note is to apply the rule to every term separately if the function is a sum of multiple polynomial terms.
This rule doesn’t work on logarithmic or trigonometric functions directly, so relying solely on it can be misleading in complex equations.

Product Rule

The product rule is crucial when differentiating products of two functions. Unlike a simple derivative of separate terms, the product rule allows us to handle expressions where functions are multiplied together. If you have a function such as \(f(x) \cdot g(x)\), the product rule tells you to take the derivative of the first function and multiply it by the second, then add the product of the derivative of the second function with the first function. Mathematically, it is expressed as:\[ \frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x) \]

• Understanding this rule is key to tackling more complex expressions involving multiple functions.
• It is particularly useful when derivatives involve logarithmic or exponential functions within products, as seen in functions like \(x^3 \ln(x)\).

Apply this rule one step at a time to avoid confusion, making sure each component is correctly differentiated before combining them back together. This method ensures accuracy, especially when dealing with longer, more complicated problems.