Differentiate.
[tex]f(x)=\ln \left(\frac{x^2-10}{x}\right)[/tex]
[tex]f^{\prime}(x)=[/tex]
Answer (2)
Simplify the function using logarithm properties: f ( x ) = ln ( x 2 − 10 ) − ln ( x ) .
Apply the chain rule to differentiate each term: d x d ln ( x 2 − 10 ) = x 2 − 10 2 x and d x d ln ( x ) = x 1 .
Combine the derivatives: f ′ ( x ) = x 2 − 10 2 x − x 1 .
Simplify the expression to obtain the final answer: f ′ ( x ) = x ( x 2 − 10 ) x 2 + 10 .
Explanation
Problem Analysis We are given the function f ( x ) = ln ( x x 2 − 10 ) and we need to find its derivative f ′ ( x ) .
Simplifying the Function We can simplify the function using logarithm properties: f ( x ) = ln ( x 2 − 10 ) − ln ( x ) . This makes differentiation easier.
Applying the Chain Rule Now, we differentiate f ( x ) with respect to x using the chain rule. The derivative of ln ( u ) is u 1 ⋅ d x d u . So, we have:
f ′ ( x ) = d x d [ ln ( x 2 − 10 ) − ln ( x )] = d x d ln ( x 2 − 10 ) − d x d ln ( x )
Differentiating the First Term Let's find the derivative of ln ( x 2 − 10 ) . Using the chain rule, we get:
d x d ln ( x 2 − 10 ) = x 2 − 10 1 ⋅ d x d ( x 2 − 10 ) = x 2 − 10 1 ⋅ ( 2 x ) = x 2 − 10 2 x
Differentiating the Second Term Next, we find the derivative of ln ( x ) :
d x d ln ( x ) = x 1
Combining the Derivatives Now, we combine the results:
f ′ ( x ) = x 2 − 10 2 x − x 1
Simplifying the Expression To simplify the expression, we find a common denominator:
f ′ ( x ) = x ( x 2 − 10 ) 2 x 2 − ( x 2 − 10 ) = x ( x 2 − 10 ) 2 x 2 − x 2 + 10 = x ( x 2 − 10 ) x 2 + 10
Final Answer Therefore, the derivative of the given function is:
f ′ ( x ) = x ( x 2 − 10 ) x 2 + 10
Examples
Understanding derivatives is crucial in many real-world applications. For example, if f ( x ) represents the position of a car at time x , then f ′ ( x ) represents the car's velocity. In this problem, we found the derivative of a logarithmic function, which can model various phenomena such as the intensity of sound or the pH level of a solution. Knowing how to differentiate such functions allows us to analyze rates of change in these scenarios, providing valuable insights into their behavior.
To differentiate f ( x ) = ln ( x x 2 − 10 ) , we first simplify it to find f ( x ) = ln ( x 2 − 10 ) − ln ( x ) . Applying the chain rule gives f ′ ( x ) = x 2 − 10 2 x − x 1 , which simplifies to f ′ ( x ) = x ( x 2 − 10 ) x 2 + 10 .
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