Zipzap said:Ok, I am really stuck on this one. I tried using the first principles formula and everything, but I don't get two separate terms like I am supposed to in the actual derivative. I always end up with -1/(3x-1)^2 when I try. Can someone please help me out?
Deriving a function using first principles is a method of finding the derivative of a function by using the definition of a derivative. In this case, we are finding the derivative of the function f(x) = x/(3x-1).
The derivative of a function f(x) is the rate of change of the function at a specific point, represented by f'(x) or dy/dx. It is defined as the limit of the difference quotient as the change in the independent variable approaches zero.
The difference quotient is a mathematical expression that represents the slope of a secant line between two points on a curve. It can also be used to find the derivative of a function by taking the limit as the change in the independent variable approaches zero.
To find the derivative of a function using first principles, we use the definition of a derivative and the difference quotient. We take the limit of the difference quotient as the change in the independent variable approaches zero. This will give us the slope of the tangent line at a specific point, which is the derivative of the function at that point.
Step 1: Write the difference quotient for the function f(x) = x/(3x-1).
Step 2: Simplify the difference quotient by expanding the numerator and denominator.
Step 3: Take the limit as h (change in independent variable) approaches zero.
Step 4: Substitute 0 for h in the simplified difference quotient.
Step 5: Simplify the resulting expression, which will give us the derivative of f(x).
Step 6: Write the final answer as f'(x) or dy/dx.