Question

For f'(e³), Let f(x)= ln(x²) Find: f'(x) = 2/x Calculate: f'(e³) = ?

In this question, you are asked to find the derivative of the function f(x), and then specifically evaluate the derivative at the point X = e³, denoted as f'(e³). Please perform this calculation.

Answer 1

To calculate f'(e^3) for the function f(x)= ln(x^2), substitute the value e^3 into the derivative f'(x) = 2/x, obtaining f'(e^3) = 2/e^3, which simplifies to 2e^-3.

Step-by-step explanation:

Calculating the Derivative Value at a Specific Point

To solve for f'(e^3), first we understand that the given function f(x) = ln(x^2) has been differentiated to f'(x) = 2/x. This is based on the chain rule for derivatives where the derivative of ln(x) is 1/x, and considering the derivative of the inner function x^2, which is 2x. To find the derivative at a specific point, we simply substitute the value x = e^3 into the derivative formula.

f'(e^3) = 2 / e^3

By computing this, we are applying the concept that the natural logarithm (In) is the power to which e, the base of the natural logarithm, must be raised to equal the number. Due to the relationship between exponential and logarithmic functions, we are utilizing these properties to evaluate the growth rate at the point e^3. The function and its derivative allow us to see how the logarithmic behavior changes at this specific value.

The final answer for f'(e^3) is 2 / e^3 which simplifies to 2e^-3, showing us the rate of change of the function at e^3.