Use logarithmic differentiation to find the derivative of y with respect to the independent variable.
-$$y = ( 10 x + 2 ) ^ { x }$$
A) $x \ln ( 10 x + 2 )$
B) $( 10 x + 2 ) ^ { x } \left( \ln ( 10 x + 2 ) + \frac { 1 } { 2 } \right)$
C) $\ln ( 10 x + 2 ) + \frac { 10 x } { 10 x + 2 }$
D) $( 10 x + 2 ) ^ { x } \left( \ln ( 10 x + 2 ) + \frac { 10 x } { 10 x + 2 } \right)$

Question

Quizplus · Accepted Answer

1. Take the natural logarithm of both sides of the equation:
$$\ln y = \ln ((10x+2)^x)$$

2. Use the power rule of logarithms to simplify the expression on the right-hand side:
$$\ln y = x \ln (10x+2)$$

3. Differentiate both sides with respect to x using the chain rule and product rule:
$$\frac{1}{y} \frac{dy}{dx} = \ln (10x+2) + x \cdot \frac{1}{10x+2} \cdot 10$$

4. Multiply both sides by y and simplify:
$$\frac{dy}{dx} = y \left(\ln (10x+2) + \frac{10x}{10x+2}\right)$$
$$= (10x+2)^x \left(\ln (10x+2) + \frac{10x}{10x+2}\right)$$

Therefore, the answer is D.

Use Logarithmic Differentiation to Find the Derivative of Y with Respect