Differentiate.
[tex]f(x)=x^6 \ln 7 x[/tex]
[tex]f^{\prime}(x)=[/tex]
(Use parentheses to clearly denote the argument of each function.)
Answer (2)
Apply the product rule: f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) , where u ( x ) = x 6 and v ( x ) = ln ( 7 x ) .
Find the derivatives: u ′ ( x ) = 6 x 5 and v ′ ( x ) = x 1 .
Substitute into the product rule: f ′ ( x ) = 6 x 5 ln ( 7 x ) + x 6 ⋅ x 1 .
Simplify: f ′ ( x ) = x 5 ( 6 ln ( 7 x ) + 1 ) . The final answer is x 5 ( 6 ln ( 7 x ) + 1 ) .
Explanation
Problem Analysis We are given the function f ( x ) = x 6 ln ( 7 x ) and asked to find its derivative f ′ ( x ) . This requires the product rule and the chain rule.
Applying the Product Rule The product rule states that if f ( x ) = u ( x ) v ( x ) , then f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) . In our case, let u ( x ) = x 6 and v ( x ) = ln ( 7 x ) .
Differentiating u(x) First, we find the derivative of u ( x ) = x 6 . Using the power rule, we have u ′ ( x ) = 6 x 5 .
Differentiating v(x) Next, we find the derivative of v ( x ) = ln ( 7 x ) . Using the chain rule, we have v ′ ( x ) = 7 x 1 ⋅ d x d ( 7 x ) = 7 x 1 ⋅ 7 = x 1 .
Applying the Product Rule Formula Now, we apply the product rule: f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) = 6 x 5 ln ( 7 x ) + x 6 ⋅ x 1 .
Simplifying the Expression Simplifying the expression, we get f ′ ( x ) = 6 x 5 ln ( 7 x ) + x 5 .
Factoring the Expression We can factor out x 5 to get f ′ ( x ) = x 5 ( 6 ln ( 7 x ) + 1 ) .
Final Answer Therefore, the derivative of f ( x ) = x 6 ln ( 7 x ) is f ′ ( x ) = x 5 ( 6 ln ( 7 x ) + 1 ) .
Examples
In economics, if you have a cost function that depends on a variable x (e.g., the number of units produced) in the form C ( x ) = x 6 ln ( 7 x ) , finding the derivative C ′ ( x ) gives you the marginal cost. The marginal cost represents the rate of change of the cost with respect to the number of units produced, which is a crucial concept for optimizing production and pricing strategies.
To differentiate f ( x ) = x 6 ln ( 7 x ) , we use the product rule which results in f ′ ( x ) = x 5 ( 6 ln ( 7 x ) + 1 ) .
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