Differentiate the function.
f(x)=3x ln 6x - 3x
f'(x) = 3x(6/6x) + 3ln(6x) - 3
= 3 - 3ln(6x) - 3
= -3ln(6x)
To differentiate the function f(x) = 3x ln(6x) - 3x, you can use the rules of differentiation. Let's break it down step by step:
Step 1: Apply the Product Rule
The Product Rule is used when you have two functions multiplied together. It states that if you have two functions f(x) and g(x), then the derivative of their product, f(x)g(x), is given by the formula:
(fg)'(x) = f'(x)g(x) + f(x)g'(x)
In our case, f(x) = 3x and g(x) = ln(6x).
Step 2: Differentiate f(x) = 3x
The derivative of f(x) = 3x with respect to x can be obtained by multiplying the coefficient 3 by the power of x, which is 1:
f'(x) = 3
Step 3: Differentiate g(x) = ln(6x)
To differentiate ln(6x), you will use the Chain Rule. The Chain Rule states that if you have a composition of functions, f(g(x)), then the derivative of f(g(x)) with respect to x is given by the formula:
(f(g(x)))' = f'(g(x)) * g'(x)
In our case, f(u) = ln(u) and g(x) = 6x. Let's find the derivatives of these functions:
Derivative of f(u) = ln(u) with respect to u:
f'(u) = 1/u
Derivative of g(x) = 6x with respect to x:
g'(x) = 6
Now, apply the Chain Rule by substituting the derivatives of f(u) and g(x) into the formula:
(ln(6x))' = (1/(6x)) * 6 = 1/x
Step 4: Combine the results
Using the Product Rule formula from Step 1, we have:
(fg)'(x) = f'(x)g(x) + f(x)g'(x)
= (3)(ln(6x)) + (3x)(1/x)
= 3ln(6x) + 3
So, the derivative of f(x) = 3x ln(6x) - 3x is 3ln(6x) + 3.