If f(x) =6ln(6x+3ln(x))
How do I Find the first derivative
use the chain rule.
If y = f(u) and u is a function of x, then
dy/dx = dy/du * du/dx
so, in this case,
f = 6lnu
u = 6x + 3lnx
f'(x) = 6/u * du/dx = 6/(6x + 3lnx) * (6 + 3/x) = 54(2x + lnx)(2 + 1/x)
To find the first derivative of f(x) = 6ln(6x + 3ln(x)), we can use the chain rule and the derivative of the natural logarithm.
Here is the step-by-step solution:
Step 1: Identify the composite functions within the expression. In this case, we have 6x + 3ln(x) inside the natural logarithm function.
Step 2: Apply the chain rule. The chain rule states that if we have a composite function f(g(x)), then the derivative is given by f'(g(x)) * g'(x).
Step 3: Find the derivative of the natural logarithm. The derivative of ln(x) with respect to x is 1/x.
Step 4: Find the derivative of the composite function (6x + 3ln(x)). To do this, we need to apply the chain rule. The derivative of 6x is 6, and the derivative of 3ln(x) is 3/x.
Step 5: Combine the results from steps 3 and 4. Using the chain rule, we multiply the derivative of the outer function (6ln) by the derivative of the inner function (6 + 3/x).
Step 6: Simplify the expression. Multiply 6 by 6 to get 36 and combine it with 3/x.
Therefore, the first derivative of f(x) = 6ln(6x + 3ln(x)) is f'(x) = 36 + 3/x.
To find the first derivative of the function f(x) = 6ln(6x+3ln(x)), you can use the chain rule and the derivative of the natural logarithm function.
Here's a step-by-step process to find the first derivative:
1. Rewrite the function using the properties of logarithms:
f(x) = 6ln(6x+ln(x^3))
2. Apply the chain rule, which states that if we have a composition of functions f(g(x)), then the derivative is given by f'(g(x)) * g'(x).
3. Let u = 6x + ln(x^3). Therefore, g(x) = u and f(u) = 6ln(u).
4. Find the derivative of g(x):
g'(x) = d/dx(6x + ln(x^3))
= 6 + (1/x^3) * 3x^2
5. Find the derivative of f(u) with respect to u:
f'(u) = d/du(6ln(u))
= (6/u)
6. Use the chain rule to find the derivative of f(x):
f'(x) = f'(u) * g'(x)
= (6/u) * (6 + (1/x^3) * 3x^2)
7. Substitute u back in:
f'(x) = (6/(6x + ln(x^3))) * (6 + (1/x^3) * 3x^2)
Therefore, the first derivative of the function f(x) = 6ln(6x + 3ln(x)) is given by:
f'(x) = (6/(6x + ln(x^3))) * (6 + (1/x^3) * 3x^2)