Find the derivative of the function.
f(x) = ln((x6 - 7)6)
To find the derivative of the function f(x) = ln((x^6 - 7)^6), we can make use of the chain rule. The chain rule states that if we have a composition of functions, the derivative of the composition can be found by multiplying the derivative of the outer function with the derivative of the inner function.
Let's break down the function into its composed functions:
Outer function: ln(u), where u = (x^6 - 7)^6
Inner function: u = (x^6 - 7)^6
Now, let's find the derivatives step by step:
Step 1: Find the derivative of the inner function with respect to x.
To do this, we need to apply the chain rule. Let v(x) = x^6 - 7, then u(x) = (v(x))^6.
Using the chain rule, the derivative of u(x) with respect to x is du/dx = (dv/dx) * (du/dv).
dv/dx = 6x^(6-1) - 0 = 6x^5
du/dv = 6(v)^(6-1) = 6(v)^5 = 6(x^6 - 7)^5
Therefore, du/dx = (dv/dx) * (du/dv) = (6x^5) * 6(x^6 - 7)^5
Step 2: Find the derivative of the outer function with respect to u.
The derivative of ln(u) with respect to u is 1/u.
Step 3: Combine the derivatives using the chain rule.
According to the chain rule, the derivative of f(x) = ln((x^6 - 7)^6) is given by df/dx = (du/dx) * (1/u).
Substituting the values we found in Step 1 and Step 2, we have:
df/dx = (6x^5) * 6(x^6 - 7)^5 * (1/(x^6 - 7)^6)
Simplifying the expression further, we get:
df/dx = 36x^5(x^6 - 7)^5 / (x^6 - 7)^6
Therefore, the derivative of the function f(x) = ln((x^6 - 7)^6) is df/dx = 36x^5(x^6 - 7)^5 / (x^6 - 7)^6.