Final Answer:
The derivative of the function f(x) = {11}√ [3]{x²+ 4 - √{x-2}} can be found using the chain rule and the power rule for differentiation. The derivative f'(x) = 11/2√[3](x² + 4 - √{x-2}) * (2x + 1/2(x-2)^(-1/2)).
Step-by-step explanation:
To find the derivative of the given function f(x) = {11}√ [3]{x²+ 4 - √{x-2}}, we use the chain rule and power rule for differentiation. First, we identify the outermost function as the cube root {11}√[3]{}, and the inner function as x²+ 4 - √{x-2}. Differentiating the outer function with respect to the inner function yields 11/2√[3](x²+ 4 - √{x-2})^(2/3-1). Then, applying the chain rule, we multiply this result by the derivative of the inner function, which is 2x + 1/2(x-2)^(-1/2).
The chain rule enables us to differentiate composite functions, where the derivative of an outer function depends on the derivative of an inner function. In this case, the function involves nested radicals and a combination of functions within one another. By applying the chain rule and power rule, we sequentially differentiate the outer and inner functions, arriving at the derivative f'(x) = 11/2√[3](x² + 4 - √{x-2}) * (2x + 1/2(x-2)^(-1/2)). This derivative represents the rate of change of the function f(x) with respect to x at any given point.