Differentiate.
$f(x)=7 \ln x$
$f^{\prime}(x) =$
Answer (1)
Apply the constant multiple rule: f ′ ( x ) = 7 d x d ln x .
Use the derivative of the natural logarithm: d x d ln x = x 1 .
Combine the results to get the derivative: f ′ ( x ) = 7 ⋅ x 1 .
The derivative of f ( x ) = 7 ln x is x 7 .
Explanation
Problem Analysis We are given the function f ( x ) = 7 x and asked to find its derivative, f ′ ( x ) .
Derivative Rules To find the derivative of f ( x ) = 7 x , we'll use the constant multiple rule and the derivative of the natural logarithm function. The constant multiple rule states that if f ( x ) = c g ( x ) , where c is a constant, then f ′ ( x ) = c g ′ ( x ) . The derivative of the natural logarithm function is d x d ( ln x ) = x 1 .
Applying the Rules Applying the constant multiple rule, we have
f ′ ( x ) = 7 d x d ( ln x )
Since d x d ( ln x ) = x 1 , we get
f ′ ( x ) = 7 ⋅ x 1 = x 7
Final Answer Therefore, the derivative of f ( x ) = 7 x is f ′ ( x ) = x 7 .
Examples
In finance, if you have an investment whose value grows logarithmically with time, such as V ( t ) = 7 ln t , finding the derivative d t d V = t 7 tells you the instantaneous rate of growth of the investment at any given time t . This is useful for making informed decisions about when to buy or sell assets.
