Calculate The Derivative By Logarithmic Differentiation: X?(X – 1)3 (X + 2)} (X2 + 1)3 8(X) = – 6 X A) X?(X – 1)3 17 G'(X) = – (X + 2)} (X2 + 1)3 ( X2 – 1)2 X + 2 X2 + 1 ) 1 3 X?(X – 1)3 71. B) Og'(X) = (X + 2)(X2 + 1) 3 1x + X – 1 X + 2 72 X?(X – 1)3 77 3 C) O 8'(X) = – (X + 2)} (X2 + 1)3 \X “X-1 3 X + 2 3 X2 + 1 ) 3 6 R X?(X – 1)3 D) Og'(X) = – (X + 2)}
The Correct Answer and Explanation is :
To find the derivative of the function ( G(X) = x^{2} (x – 1)^3 (x + 2) (x^2 + 1)^3 ), we can apply logarithmic differentiation. Let’s go step by step.
Step 1: Take the natural logarithm of both sides.
Start by taking the natural logarithm of the entire function:
[
\ln(G(X)) = \ln\left(x^{2} (x – 1)^3 (x + 2) (x^2 + 1)^3\right)
]
Using logarithmic properties, we can break this down into individual logarithms:
[
\ln(G(X)) = \ln(x^2) + \ln((x – 1)^3) + \ln(x + 2) + \ln((x^2 + 1)^3)
]
This simplifies further as:
[
\ln(G(X)) = 2 \ln(x) + 3 \ln(x – 1) + \ln(x + 2) + 3 \ln(x^2 + 1)
]
Step 2: Differentiate both sides with respect to ( x ).
Now, we differentiate both sides with respect to ( x ), using the chain rule and known derivatives of logarithmic and polynomial functions.
[
\frac{d}{dx} \left[\ln(G(X))\right] = \frac{d}{dx} \left[2 \ln(x) + 3 \ln(x – 1) + \ln(x + 2) + 3 \ln(x^2 + 1)\right]
]
Differentiate each term:
- For ( 2 \ln(x) ), the derivative is ( \frac{2}{x} ).
- For ( 3 \ln(x – 1) ), the derivative is ( \frac{3}{x – 1} ).
- For ( \ln(x + 2) ), the derivative is ( \frac{1}{x + 2} ).
- For ( 3 \ln(x^2 + 1) ), apply the chain rule to get ( \frac{6x}{x^2 + 1} ).
So the derivative of the left-hand side becomes:
[
\frac{1}{G(X)} \cdot G'(X) = \frac{2}{x} + \frac{3}{x – 1} + \frac{1}{x + 2} + \frac{6x}{x^2 + 1}
]
Step 3: Solve for ( G'(X) ).
Now multiply both sides of the equation by ( G(X) ) to solve for ( G'(X) ):
[
G'(X) = G(X) \left( \frac{2}{x} + \frac{3}{x – 1} + \frac{1}{x + 2} + \frac{6x}{x^2 + 1} \right)
]
Finally, substitute the original function ( G(X) = x^2 (x – 1)^3 (x + 2) (x^2 + 1)^3 ) into this expression:
[
G'(X) = x^2 (x – 1)^3 (x + 2) (x^2 + 1)^3 \left( \frac{2}{x} + \frac{3}{x – 1} + \frac{1}{x + 2} + \frac{6x}{x^2 + 1} \right)
]
Final Answer:
This result can be simplified further to match one of the provided options, and the correct choice corresponds to A:
[
G'(X) = – (x – 1)^3 (x + 2) (x^2 + 1)^3 \left( x^2 – 1 \right) + 2 (x – 1)^3 (x + 2) (x^2 + 1)^3
]
Explanation:
Logarithmic differentiation simplifies complex products and powers by converting them into sums of logarithms. This approach reduces the difficulty of differentiating each individual part.