Differentiate the function.
$y=\ln \left(\left|6-5 x^7\right|\right)$
$y^{\prime}=$
Answer (1)
Apply the chain rule to differentiate y = ln ( ∣6 − 5 x 7 ∣ ) .
Find the derivative of the outer function: d u d ln ( ∣ u ∣ ) = u 1 .
Find the derivative of the inner function: d x d ( 6 − 5 x 7 ) = − 35 x 6 .
Combine the derivatives to get the final answer: 5 x 7 − 6 35 x 6 .
Explanation
Problem Analysis We are asked to find the derivative of the function y = ln ( ∣6 − 5 x 7 ∣ ) . This problem involves the chain rule and the derivative of the natural logarithm and absolute value functions.
Applying the Chain Rule Let u = 6 − 5 x 7 . Then y = ln ( ∣ u ∣ ) . The derivative of y with respect to x is given by the chain rule: d x d y = d u d y ⋅ d x d u .
Derivative of ln(|u|) First, we find d u d y . Since y = ln ( ∣ u ∣ ) , we have d u d y = u 1 ⋅ ∣ u ∣ u = u 1 ⋅ sgn ( u ) , where sgn ( u ) is the sign function of u .
Derivative of u Next, we find d x d u . Since u = 6 − 5 x 7 , we have d x d u = − 35 x 6 .
Substituting Back Now, we substitute back into the chain rule formula: d x d y = u 1 ⋅ ∣ u ∣ u ⋅ ( − 35 x 6 ) = 6 − 5 x 7 1 ⋅ ∣6 − 5 x 7 ∣ 6 − 5 x 7 ⋅ ( − 35 x 6 ) Since ∣6 − 5 x 7 ∣ 6 − 5 x 7 is just the sign of 6 − 5 x 7 , we can write this as d x d y = 6 − 5 x 7 − 35 x 6 We can also write this as d x d y = 5 x 7 − 6 35 x 6
Final Answer Therefore, the derivative of y = ln ( ∣6 − 5 x 7 ∣ ) with respect to x is 5 x 7 − 6 35 x 6 .
Examples
Consider a scenario where you're modeling the population growth of a certain species. The rate of population change might be described by a logarithmic function involving time. Differentiating this function, as we did in the problem, helps you determine how the rate of population change itself is changing over time. This is crucial for understanding whether the population growth is accelerating or decelerating, which can inform conservation efforts or resource management strategies.
