Use logarithmic differentiation to find the derivative of y.
-$$y = \left( x ^ { 3 } + 1 \right) ^ { 4 } ( x - 1 ) ^ { 5 } x ^ { 3 }$$
A) $\left( x ^ { 3 } + 1 \right) ^ { 4 } ( x - 1 ) ^ { 5 } x ^ { 3 } \left( 4 \ln \left( x ^ { 3 } + 1 \right) + 5 \ln ( x - 1 ) + 3 \ln x \right)$
B) $\left( x ^ { 3 } + 1 \right) ^ { 4 } ( x - 1 ) ^ { 5 } x ^ { 3 } \left( \frac { 15 } { x } + \frac { 5 } { x - 1 } \right)$
C) $\left( x ^ { 3 } + 1 \right) ^ { 4 } ( x - 1 ) ^ { 5 } x ^ { 3 } \left( \frac { 12 x ^ { 2 } } { x ^ { 3 } + 1 } + \frac { 5 } { x - 1 } + \frac { 3 } { x } \right)$
D) $\frac { 12 x ^ { 2 } } { x ^ { 3 } + 1 } + \frac { 5 } { x - 1 } + \frac { 3 } { x }$

Question

Quizplus · Accepted Answer

1. Take the natural logarithm of both sides:
$$\ln y = 4 \ln (x^3+1) + 5 \ln (x-1) + 3 \ln x $$

2. Differentiate both sides with respect to x using the chain rule on the right-hand side:
$$\frac{1}{y} \frac{dy}{dx} = \frac{12x^2}{x^3+1} + \frac{5}{x-1} + \frac{3}{x}$$

3. Multiply both sides by y:
$$\frac{dy}{dx} = y \left(\frac{12x^2}{x^3+1} + \frac{5}{x-1} + \frac{3}{x} \right)$$

4. Substitute the expression for y:
$$\frac{dy}{dx} = \left( x ^ { 3 } + 1 \right) ^ { 4 } ( x - 1 ) ^ { 5 } x ^ { 3 } \left(\frac{12x^2}{x^3+1} + \frac{5}{x-1} + \frac{3}{x} \right)$$

Simplifying this expression yields the answer choice C.

Use Logarithmic Differentiation to Find the Derivative of Y y=(x3+1)4(x−1)5x3y = \left( x ^ { 3 } + 1 \right) ^ { 4 } ( x - 1 ) ^ { 5 } x ^ { 3 }y=(x3+1)4(x−1)5x3

Use Logarithmic Differentiation to Find the Derivative of Y $y = \left( x ^ { 3 } + 1 \right) ^ { 4 } ( x - 1 ) ^ { 5 } x ^ { 3 }$