Final answer:
To differentiate the given functions, the quotient rule and the product rule are applied. For the first function, the quotient rule is used to find the derivative, and for the second function, the product rule is employed. These rules are essential in calculus for finding rates of change for expressions involving division or multiplication of functions.
Step-by-step explanation:
The student has asked to differentiate two functions:
- y = √(3x - 4) / (5x + 3), which can also be written as y = (3x - 4)1/2 / (5x + 3)
- y = (3x - 4)1/2 (5x + 3)
To differentiate these functions, we'll use the quotient rule and the product rule of differentiation, respectively.
The quotient rule is given by dx² - y² / dxz, and the product rule states that the derivative of two functions multiplied together is the derivative of the first function times the second function plus the derivative of the second function times the first function.
For the first function, using the quotient rule:
Let u = (3x - 4)1/2 and v = (5x + 3). The derivative du/dx = (1/2)(3x - 4)-1/2 × 3 and dv/dx = 5. Thus, the derivative dy/dx = [(5)(3x - 4)1/2 - (5x + 3)(3/2)(3x - 4)-1/2 × 3] / (5x + 3)2.
For the second function, using the product rule:
The derivative dy/dx = (3/2)(3x - 4)-1/2 × (5x + 3) + (3x - 4)1/2 × 5.
Note: The expression y* button or raising to the power is a hint for calculating powers of expressions. Raising to the power 0.25 calculates the fourth root, analogous to raising to the power 0.5 to calculate the square root.