Final answer:
The derivative of f(x) = 4x^(5/4) + 6x^(3/2) is f'(x) = 5x^(1/4) + 9x^(1/2). f'(1) = 14 and f'(81) = 96.
Step-by-step explanation:
To find the derivative of the function f(x) = 4x^(5/4) + 6x^(3/2), we will use the power rule and the sum rule of differentiation.
Applying the power rule, the derivative of x^n is n*x^(n-1), the derivative of 4x^(5/4) is 5/4 * 4 * x^(5/4 - 1) = 5x^(1/4). Applying the power rule again, the derivative of 6x^(3/2) is 3/2 * 6 * x^(3/2 - 1) = 9x^(1/2).
Therefore, the derivative of f(x) = 4x^(5/4) + 6x^(3/2) is f'(x) = 5x^(1/4) + 9x^(1/2).
Now, we can find f'(1) by substituting x = 1 into f'(x). f'(1) = 5(1)^(1/4) + 9(1)^(1/2) = 5 + 9 = 14.
Similarly, to find f'(81), we substitute x = 81 into f'(x). f'(81) = 5(81)^(1/4) + 9(81)^(1/2) = 5 * 3 + 9 * 9 = 15 + 81 = 96.