Question

If y = (x - 1)² (x - 2)³ (x - 3)⁵ then \(\frac{dy}{dx}\) at x = 4 is equal to

• A. 108
• B. 54
• C. 36
• D. 516

Answer 1

The Correct Option is D

We are asked to find the value of the derivative \( \frac{dy}{dx} \) at \( x = 4 \) for the function:

\[ y = (x - 1)^2 (x - 2)^3 (x - 3)^5 \]

This function is a product of powers, so logarithmic differentiation is a convenient method.

Step 1: Take the natural logarithm of both sides.

\[ \ln y = \ln \left[ (x - 1)^2 (x - 2)^3 (x - 3)^5 \right] \]

Using logarithm properties \( \ln(abc) = \ln a + \ln b + \ln c \) and \( \ln(a^p) = p \ln a \):

\[ \ln y = \ln(x - 1)^2 + \ln(x - 2)^3 + \ln(x - 3)^5 \] \[ \ln y = 2 \ln(x - 1) + 3 \ln(x - 2) + 5 \ln(x - 3) \]

Step 2: Differentiate both sides with respect to \( x \).

Using implicit differentiation for \( \ln y \) and the chain rule for the terms on the right:

\[ \frac{d}{dx}(\ln y) = \frac{d}{dx} [ 2 \ln(x - 1) + 3 \ln(x - 2) + 5 \ln(x - 3) ] \] \[ \frac{1}{y} \frac{dy}{dx} = 2 \cdot \frac{1}{x - 1} \cdot (1) + 3 \cdot \frac{1}{x - 2} \cdot (1) + 5 \cdot \frac{1}{x - 3} \cdot (1) \] \[ \frac{1}{y} \frac{dy}{dx} = \frac{2}{x - 1} + \frac{3}{x - 2} + \frac{5}{x - 3} \]

Step 3: Solve for \( \frac{dy}{dx} \).

\[ \frac{dy}{dx} = y \left[ \frac{2}{x - 1} + \frac{3}{x - 2} + \frac{5}{x - 3} \right] \]

Substitute the original expression for \( y \):

\[ \frac{dy}{dx} = (x - 1)^2 (x - 2)^3 (x - 3)^5 \left[ \frac{2}{x - 1} + \frac{3}{x - 2} + \frac{5}{x - 3} \right] \]

Step 4: Evaluate the derivative at \( x = 4 \).

First, evaluate \( y \) at \( x = 4 \):

\[ y(4) = (4 - 1)^2 (4 - 2)^3 (4 - 3)^5 = (3)^2 (2)^3 (1)^5 = 9 \times 8 \times 1 = 72 \]

Next, evaluate the expression in the brackets at \( x = 4 \):

\[ \left[ \frac{2}{x - 1} + \frac{3}{x - 2} + \frac{5}{x - 3} \right]_{x=4} = \frac{2}{4 - 1} + \frac{3}{4 - 2} + \frac{5}{4 - 3} \] \[ = \frac{2}{3} + \frac{3}{2} + \frac{5}{1} \] \[ = \frac{2 \times 2}{6} + \frac{3 \times 3}{6} + \frac{5 \times 6}{6} \] \[ = \frac{4 + 9 + 30}{6} = \frac{43}{6} \]

Now, substitute these values back into the expression for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx}\Bigg|_{x=4} = y(4) \times \left( \frac{43}{6} \right) \] \[ = 72 \times \frac{43}{6} \] \[ = (12 \times 6) \times \frac{43}{6} \] \[ = 12 \times 43 \] \[ = 516 \]

The value of the derivative at \( x = 4 \) is 516.

So, the correct answer ia (D): 516

Answer 2

Given \( y = (x - 1)^2 (x - 2)^3 (x - 3)^5 \), we want to find \( \frac{dy}{dx} \) at \( x = 4 \).

First, we take the natural logarithm of both sides:

\[ \ln y = \ln \left[ (x - 1)^2 (x - 2)^3 (x - 3)^5 \right] \]

\[ \ln y = 2 \ln(x - 1) + 3 \ln(x - 2) + 5 \ln(x - 3) \]

Now, differentiate both sides with respect to \( x \):

\[ \frac{1}{y} \frac{dy}{dx} = \frac{2}{x - 1} + \frac{3}{x - 2} + \frac{5}{x - 3} \]

Thus, the derivative is:

\[ \frac{dy}{dx} = y \left( \frac{2}{x - 1} + \frac{3}{x - 2} + \frac{5}{x - 3} \right) \]

Now, substitute \( x = 4 \) into the equation:

\[ y(4) = (4 - 1)^2 (4 - 2)^3 (4 - 3)^5 = (3)^2 (2)^3 (1)^5 = 9 \cdot 8 \cdot 1 = 72 \]

Substituting \( x = 4 \) into the derivative formula:

\[ \frac{dy}{dx} \bigg|_{x=4} = 72 \left( \frac{2}{4 - 1} + \frac{3}{4 - 2} + \frac{5}{4 - 3} \right) \]

\[ \frac{dy}{dx} \bigg|_{x=4} = 72 \left( \frac{2}{3} + \frac{3}{2} + \frac{5}{1} \right) \]

\[ \frac{dy}{dx} \bigg|_{x=4} = 72 \left( \frac{4 + 9 + 30}{6} \right) = 72 \times \frac{43}{6} = 12 \times 43 = 516 \]

Therefore, \( \frac{dy}{dx} \) at \( x = 4 \) is equal to 516.