Use logarithmic differentiation to find the derivative of y with respect to the independent variable.
-$$y = ( x + 3 ) ^ { x }$$
A) $x + ( 3 ) ^ { x - 1 }$
B) $\ln ( x + 3 ) + \frac { x } { x + 3 }$
C) $( x + 3 ) ^ { x } \left( \ln ( x + 3 ) + \frac { x } { x + 3 } \right)$
D) $x \ln ( x + 3 )$

Question

Quizplus · Accepted Answer

We take the natural logarithm of both sides:
$$\ln y = x \ln (x + 3)$$
Now we differentiate both sides implicitly with respect to x:
$$\frac{1}{y} \cdot\frac{dy}{dx} = \ln (x + 3) + x \cdot\frac{1}{x + 3}$$
Multiplying both sides by y:
$$\frac{dy}{dx} = y \left(\ln (x + 3) + \frac{x}{x+3}\right)$$
Substituting the expression for y:
$$\frac{dy}{dx} = ( x + 3 ) ^ { x } \left( \ln ( x + 3 ) + \frac { x } { x + 3 } \right)$$ 
Therefore, the answer is C.

Use Logarithmic Differentiation to Find the Derivative of Y with Respect