Question

Let f(x)=( x+3
x+6
​
) 9
f ′
(x)=

Answer 1

Explanation:

To find the derivative of the function f(x) = ((x+3)/(x+6))^9, we can use the chain rule and the power rule for differentiation.

Let's break down the process step by step:

Step 1: Apply the chain rule by differentiating the outer function, leaving the inner function unchanged.

f'(x) = 9((x+3)/(x+6))^(9-1) * (d/dx)((x+3)/(x+6))

Step 2: Differentiate the inner function using the quotient rule.

d/dx((x+3)/(x+6)) = [(x+6)(1) - (x+3)(1)] / (x+6)^2

= (x+6 - x - 3) / (x+6)^2

= 3 / (x+6)^2

Step 3: Substitute the result from step 2 into step 1.

f'(x) = 9((x+3)/(x+6))^(9-1) * (3 / (x+6)^2)

Simplifying further:

f'(x) = 9((x+3)/(x+6))^8 * (3 / (x+6)^2)

Therefore, the derivative of the function f(x) = ((x+3)/(x+6))^9 is f'(x) = 9((x+3)/(x+6))^8 * (3 / (x+6)^2).