Find the derivative of f(x) = ln(48e3x − 11)
To find the derivative of f(x) = ln(48e3x − 11), we can use the chain rule. The chain rule states that if we have a composition of functions, such as f(g(x)), then the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function.
Let's apply the chain rule to find the derivative of f(x):
First, let's identify the inner function and the outer function:
- The outer function is ln(u), where u = 48e3x − 11.
- The inner function is u = 48e3x − 11.
Now, let's find the derivatives of the outer and inner functions:
The derivative of ln(u) with respect to u is 1/u.
The derivative of u with respect to x is du/dx = d(48e3x − 11)/dx = 48e3.
To find the derivative of f(x), we multiply the derivatives of the outer and inner functions:
df/dx = (1/u) * (du/dx)
= (1/(48e3x − 11)) * (48e3)
Therefore, the derivative of f(x) = ln(48e3x − 11) is df/dx = 48e3 / (48e3x − 11).