Find the derivative of y with respect to x.
y = ln 7x
A. 1 7x
B. 1 7x
C. 1 X
D. 1 X
The Correct Answer and Explanation is :
The correct answer is C. ( \frac{1}{x} ).
To find the derivative of ( y = \ln(7x) ), we’ll use the chain rule. The chain rule states that the derivative of a composite function ( f(g(x)) ) is ( f'(g(x)) \cdot g'(x) ).
In the given function, let:
- ( u = 7x )
- ( y = \ln(u) )
First, we differentiate ( u ) with respect to ( x ):
[ u’ = \frac{d}{dx}(7x) = 7 ]
Next, we differentiate ( y ) with respect to ( u ) (since ( y = \ln(u) )):
[ y’ = \frac{d}{du}(\ln(u)) = \frac{1}{u} ]
Using the chain rule, we multiply the derivatives:
[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{1}{u} \cdot 7 = \frac{7}{7x} ]
Since ( 7 ) in the numerator and denominator cancel out, we simplify this to:
[ \frac{dy}{dx} = \frac{1}{x} ]
This means that the derivative of ( y = \ln(7x) ) with respect to ( x ) is ( \frac{1}{x} ). The factor of ( 7 ) in ( 7x ) is just a constant multiple in the argument of the logarithm, which does not affect the derivative beyond its appearance in the chain rule and then canceling out. Thus, we derive the derivative as simply ( \frac{1}{x} ), which corresponds to option C.
This differentiation process showcases the use of the chain rule in calculus, essential for tackling derivatives of composite functions like logarithms of linear expressions. Understanding and applying these principles is crucial for effective problem-solving in calculus.