Question

Derivatives of Exponential and Logarithmic Functions:

Let f(x) = -3 * log(6x). Find f(x) and f(2).

Answer 1

The derivative of the function f(x) is -3/(x log(10)), and f(2) is evaluated using the base conversion between natural and common logarithms, expressed as -3 ln(12)/ln(10), which can be approximated using a calculator.

Step-by-step explanation:

First, let's find the derivative of the function f(x) = -3 × log(6x). To do this, we'll use the properties of logarithms alongside the chain rule. Remember that when we differentiate logarithmic functions, particularly the natural logarithm, we have that the derivative of ln(x) with respect to x is 1/x. But here, we have a base-10 logarithm and a coefficient.

The derivative of log_b(u), where u is a function of x, is (1/u) × (du/dx) × (1/log(b)). So, applying this to -3 × log(6x), we get:

f'(x) = -3 × (1/(6x) × (6)) × (1/log(10))

Simplifying, we find:

f'(x) = -3/(x × log(10))

To find f(2), we substitute x = 2 into the original function:

f(2) = -3 × log(12)

Expressing '12' as e^(ln 12) using the fact that exponential and log functions are inverses, we can write:

f(2) = -3 × ln(12)/ln(10)

Using a calculator, we can find the numerical value of f(2).