To find the derivative of the function f(x) = ln(x * e^sqrt(x) + 7), we can use the chain rule and the derivative of the natural logarithm function.
Let's break down the process step by step:
1. Begin by differentiating the inner function:
- The derivative of the function inside the natural logarithm, x * e^sqrt(x) + 7, is computed by applying the product rule.
- The derivative of x is 1, and the derivative of e^sqrt(x) is e^sqrt(x) times the derivative of sqrt(x).
2. Next, differentiate the outer function:
- The derivative of ln(u), where u is a function of x, is 1/u times the derivative of u.
3. Combine the results from step 1 and step 2 using the chain rule:
- Multiply the derivative of the inner function by the derivative of the outer function.
- In this case, it will be (1/x * e^sqrt(x) + 7) times the derivative of x * e^sqrt(x) + 7.
4. Simplify the expression obtained in step 3, if possible:
- Simplify the resulting expression to obtain the final derivative of f(x).
Please note that the simplification of the expression may vary depending on the specific values and operations involved in the function.
It's important to note that finding the derivative of a function can be complex, and it's always recommended to double-check the result or consult with a teacher or textbook for confirmation.